Online math open (OMO) 2012 2017 fall, EN with solutions

Theprobability that the perpendicular bisectors of OA and OB intersect strictly inside the circle can beexpressed in the form mn, where m, n are relatively prime positive integers.. The

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The Online Math Open Fall Contest

September 24-October 1, 2012

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Each team will submit its final answers through its team account Though teams can save drafts for theiranswers, the current interface does not allow for much flexibility in communication between team members.

We recommend using Google Docs and Spreadsheets to discuss problems and compare answers, especially ifteammates cannot communicate in person Teams may spend as much time as they like on the test beforethe deadline

Aids

Drawing aids such as graph paper, ruler, and compass are permitted However, electronic drawing aids arenot allowed This is includes (but is not limited to) Geogebra and graphing calculators Published printand electronic resources are not permitted (This is a change from last year’s rules.)

Four-function calculators are permitted on the Online Math Open That is, calculators which perform onlythe four basic arithmetic operations (+-*/) may be used Any other computational aids such as scientificand graphing calculators, computer programs and applications such as Mathematica, and online databases

is prohibited All problems on the Online Math Open are solvable without a calculator Four-functioncalculators are permitted only to help participants reduce computation errors

Clarifications

Clarifications will be posted as they are answered For the Fall 2012-2013 Contest, they will be posted at here

If you have a question about a problem, please email OnlineMathOpenTeam@gmail.com with “Clarification”

in the subject We have the right to deny clarification requests that we feel we cannot answer

Scoring

Each problem will be worth one point Ties will be broken based on the highest problem number that ateam answered correctly If there are still ties, those will be broken by the second highest problem solved,and so on

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September/October 2012 Fall OMO 2012-2013 Page 3

1 Calvin was asked to evaluate 37 + 31 × a for some number a Unfortunately, his paper was tilted 45degrees, so he mistook multiplication for addition (and vice versa) and evaluated 37 × 31 + a instead.Fortunately, Calvin still arrived at the correct answer while still following the order of operations Forwhat value of a could this have happened?

2 Petya gave Vasya a number puzzle Petya chose a digit X and said, “I am thinking of a number that

is divisible by 11 The hundreds digit is X and the tens digit is 3 Find the units digit.” Vasya wasexcited because he knew how to solve this problem, but then realized that the problem Petya gave didnot have an answer What digit X did Petya chose?

3 Darwin takes an 11 × 11 grid of lattice points and connects every pair of points that are 1 unit apart,creating a 10 × 10 grid of unit squares If he never retraced any segment, what is the total length ofall segments that he drew?

4 Let lcm(a, b) denote the least common multiple of a and b Find the sum of all positive integers x suchthat x ≤ 100 and lcm(16, x) = 16x

5 Two circles have radius 5 and 26 The smaller circle passes through center of the larger one What

is the difference between the lengths of the longest and shortest chords of the larger circle that aretangent to the smaller circle?

6 An elephant writes a sequence of numbers on a board starting with 1 Each minute, it doubles thesum of all the numbers on the board so far, and without erasing anything, writes the result on theboard It stops after writing a number greater than one billion How many distinct prime factors doesthe largest number on the board have?

7 Two distinct points A and B are chosen at random from 15 points equally spaced around a circlecentered at O such that each pair of points A and B has the same probability of being chosen Theprobability that the perpendicular bisectors of OA and OB intersect strictly inside the circle can beexpressed in the form mn, where m, n are relatively prime positive integers Find m + n

8 In triangle ABC let D be the foot of the altitude from A Suppose that AD = 4, BD = 3, CD = 2,and AB is extended past B to a point E such that BE = 5 Determine the value of CE2

9 Define a sequence of integers by T1 = 2 and for n ≥ 2, Tn = 2Tn−1 Find the remainder when

T1+ T2+ · · · + T256is divided by 255

10 There are 29 unit squares in the diagram below A frog starts in one of the five (unit) squares on thetop row Each second, it hops either to the square directly below its current square (if that squareexists), or to the square down one unit and left one unit of its current square (if that square exists),until it reaches the bottom Before it reaches the bottom, it must make a hop every second How manydistinct paths (from the top row to the bottom row) can the frog take?

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11 Let ABCD be a rectangle Circles with diameters AB and CD meet at points P and Q inside therectangle such that P is closer to segment BC than Q Let M and N be the midpoints of segments

AB and CD If ∠M P N = 40◦, find the degree measure of ∠BP C

12 Let a1, a2, be a sequence defined by a1= 1 and for n ≥ 1, an+1=pa2

15 How many sequences of nonnegative integers a1, a2, , an (n ≥ 1) are there such that a1· an > 0,

19 In trapezoid ABCD, AB < CD, AB ⊥ BC, AB k CD, and the diagonals AC, BD are perpendicular

at point P There is a point Q on ray CA past A such that QD ⊥ DC If

N is divided by 1000

21 A game is played with 16 cards laid out in a row Each card has a black side and a red side, andinitially the face-up sides of the cards alternate black and red with the leftmost card black-side-up Amove consists of taking a consecutive sequence of cards (possibly only containing 1 card) with leftmost

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card black-side-up and the rest of the cards red-side-up, and flipping all of these cards over The gameends when a move can no longer be made What is the maximum possible number of moves that can

be made before the game ends?

22 Let c1, c2, , c6030 be 6030 real numbers Suppose that for any 6030 real numbers a1, a2, , a6030,there exist 6030 real numbers {b1, b2, , b6030} such that

24 In scalene 4ABC, I is the incenter, Iais the A-excenter, D is the midpoint of arc BC of the circumcircle

of ABC, and M is the midpoint of side BC Extend ray IM past M to point P such that IM = M P Let Q be the intersection of DP and M Ia, and R be the point on the line M Ia such that AR k DP Given that AIa

i = 1, 2, , 2011 can be expressed in the form m

n for relatively prime positive integers m, n, find theremainder when m + n is divided by 1000

26 Find the smallest positive integer k such that

x + kb12

≡ x12

(mod b)

for all positive integers b and x (Note: For integers a, b, c we say a ≡ b (mod c) if and only if a − b isdivisible by c.)

27 Let ABC be a triangle with circumcircle ω Let the bisector of ∠ABC meet segment AC at D andcircle ω at M 6= B The circumcircle of 4BDC meets line AB at E 6= B, and CE meets ω at P 6= C.The bisector of ∠P M C meets segment AC at Q 6= C Given that P Q = M C, determine the degreemeasure of ∠ABC

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28 Find the remainder when

216

X

k=1

2kk

(3 · 214+ 1)k(k − 1)216−1

is divided by 216+ 1 (Note: It is well-known that 216+ 1 = 65537 is prime.)

29 In the Cartesian plane, let Si,j = {(x, y) | i ≤ x ≤ j} For i = 0, 1, , 2012, color Si,i+1 pink if i iseven and gray if i is odd For a convex polygon P in the plane, let d(P ) denote its pink density, i.e.the fraction of its total area that is pink Call a polygon P pinxtreme if it lies completely in the region

S0,2013 and has at least one vertex on each of the lines x = 0 and x = 2013 Given that the minimumvalue of d(P ) over all non-degenerate convex pinxtreme polygons P in the plane can be expressed inthe form (1+

√ p)2

q 2 for positive integers p, q, find p + q

30 Let P (x) denote the polynomial

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Acknowledgments

Contest Directors

Ray Li, James Tao, Victor Wang

Head Problem Writers

Ray Li, Victor Wang

Problem Contributors

Ray Li, James Tao, Anderson Wang, Victor Wang, David Yang, Alex Zhu

Proofreaders and Test Solvers

Mitchell Lee, James Tao, Anderson Wang, David Yang, Alex Zhu

Website Manager

Ray Li

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The Online Math Open January 16-23, 2012

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Contest Information

Format

The test will start Monday January 16 and end Monday January 23 The test consists of 50 short answerquestions, each of which has a nonnegative integer answer The problem difficulties range from those of AMCproblems to those of Olympiad problems Problems are ordered in roughly increasing order of difficulty

Team Guidelines

Students may compete in teams of up to four people Participating students must not have graduated fromhigh school International students may participate No student can be a part of more than one team Themembers of each team do not get individual accounts; they will all share the team account

The team will submit their final answers through their account Though teams can save drafts for theiranswers, the current interface does not allow for much flexibility in communication between team members

Aids

Drawing aids such as graph paper, ruler, and compass are permitted However, electronic drawing aids arenot allowed This is includes (but is not limited to) Geogebra and graphing calculators Published print andelectronic resources are permitted

Four-function calculators are permitted on the Online Math Open That is, calculators which perform onlythe four basic arithmetic operations (+-*/) may be used No other computational aids such as scientificand graphing calculators, computer programs and applications such as Mathematica, and online databasesare permitted All problems on the Online Math Open are solvable without a calculator Four-functioncalculators are permitted only to help participants reduce computation errors

Scoring

Each problem will be worth one point Ties will be broken based on the highest problem number that ateam answered correctly If there are still ties, those will be broken by the second highest problem solved,and so on

Results

After the contest is over, we will release the answers to the problems within the next day If you have aprotest about an answer, you may send an email to onlinemathopen2011@gmail.com (Include ”Protest” inthe subject) Solutions and results will be released in the following weeks

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January 2012 OMO 2012 Page 3

1 The average of two positive real numbers is equal to their difference What is the ratio of the largernumber to the smaller one?

2 How many ways are there to arrange the letters A, A, A, H, H in a row so that the sequence HA appears

at least once?

3 A lucky number is a number whose digits are only 4 or 7 What is the 17th smallest lucky number?

4 How many positive even numbers have an even number of digits and are less than 10000?

5 Congruent circles Γ1 and Γ2 have radius 2012, and the center of Γ1 lies on Γ2 Suppose that Γ1 and

Γ2 intersect at A and B The line through A perpendicular to AB meets Γ1 and Γ2 again at C and

D, respectively Find the length of CD

6 Alice’s favorite number has the following properties:

• It has 8 distinct digits

• The digits are decreasing when read from left to right

• It is divisible by 180

What is Alice’s favorite number?

7 A board 64 inches long and 4 inches high is inclined so that the long side of the board makes a 30degree angle with the ground The distance from the highest point on the board to the ground can beexpressed in the form a + b√

c where a, b, c are positive integers and c is not divisible by the square ofany prime What is a + b + c?

8 An 8 × 8 × 8 cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded bythree faces of the same color The cube is then cut into 512 unit cubes How many of these cubescontain both red and blue paint on at least one of their faces?

9 At a certain grocery store, cookies may be bought in boxes of 10 or 21 What is the minimum positivenumber of cookies that must be bought so that the cookies may be split evenly among 13 people?

10 A drawer has 5 pairs of socks Three socks are chosen at random If the probability that there is apair among the three is mn, where m and n are relatively prime positive integers, what is m + n?

11 If

1

x+

12x2 + 14x3 + 18x4 + 116x5 + · · · = 1

64,and x can be expressed in the form m

n, where m, n are relatively prime positive integers, find m + n

12 A cross-pentomino is a shape that consists of a unit square and four other unit squares each sharing

a different edge with the first square If a cross-pentomino is inscribed in a circle of radius R, what is100R2?

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13 A circle ω has center O and radius r A chord BC of ω also has length r, and the tangents to ω at

B and C meet at A Ray AO meets ω at D past O, and ray OA meets the circle centered at A withradius AB at E past A Compute the degree measure of ∠DBE

14 Al told Bob that he was thinking of 2011 distinct positive integers He also told Bob the sum of thoseintegers From this information, Bob was able to determine all 2011 integers How many possible sumscould Al have told Bob?

15 Five bricklayers working together finish a job in 3 hours Working alone, each bricklayer takes at most

36 hours to finish the job What is the smallest number of minutes it could take the fastest bricklayer

to complete the job alone?

16 Let A1B1C1D1A2B2C2D2 be a unit cube, with A1B1C1D1 and A2B2C2D2opposite square faces, andlet M be the center of face A2B2C2D2 Rectangular pyramid M A1B1C1D1 is cut out of the cube Ifthe surface area of the remaining solid can be expressed in the form a +√

b, where a and b are positiveintegers and b is not divisible by the square of any prime, find a + b

17 Each pair of vertices of a regular 10-sided polygon is connected by a line segment How many unorderedpairs of distinct parallel line segments can be chosen from these segments?

18 The sum of the squares of three positive numbers is 160 One of the numbers is equal to the sum ofthe other two The difference between the smaller two numbers is 4 What is the difference betweenthe cubes of the smaller two numbers?

19 There are 20 geese numbered 1 through 20 standing in a line The even numbered geese are standing

at the front in the order 2, 4, , 20, where 2 is at the front of the line Then the odd numbered geeseare standing behind them in the order, 1, 3, 5, , 19, where 19 is at the end of the line The geesewant to rearrange themselves in order, so that they are ordered 1, 2, , 20 (1 is at the front), and they

do this by successively swapping two adjacent geese What is the minimum number of swaps required

to achieve this formation?

20 Let ABC be a right triangle with a right angle at C Two lines, one parallel to AC and the otherparallel to BC, intersect on the hypotenuse AB The lines cut the triangle into two triangles and arectangle The two triangles have areas 512 and 32 What is the area of the rectangle?

21 If

201120112012 = xxfor some positive integer x, how many positive integer factors does x have?

22 Find the largest prime number p such that when 2012! is written in base p, it has at least p trailingzeroes

23 Let ABC be an equilateral triangle with side length 1 This triangle is rotated by some angle aboutits center to form triangle DEF The intersection of ABC and DEF is an equilateral hexagon with

an area that is 4

5 the area of ABC The side length of this hexagon can be expressed in the form m

n

where m and n are relatively prime positive integers What is m + n?

24 Find the number of ordered pairs of positive integers (a, b) with a + b prime, 1 ≤ a, b ≤ 100, and ab+1

a+b

is an integer

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25 Let a, b, c be the roots of the cubic x3+ 3x2+ 5x + 7 Given that P is a cubic polynomial such that

P (a) = b + c, P (b) = c + a, P (c) = a + b, and P (a + b + c) = −16, find P (0)

26 Xavier takes a permutation of the numbers 1 through 2011 at random, where each permutation has

an equal probability of being selected He then cuts the permutation into increasing contiguous sequences, such that each subsequence is as long as possible Compute the expected number of suchsubsequences

sub-27 Let a and b be real numbers that satisfy

a4+ a2b2+ b4= 900,

a2+ ab + b2= 45

Find the value of 2ab

28 A fly is being chased by three spiders on the edges of a regular octahedron The fly has a speed of 50meters per second, while each of the spiders has a speed of r meters per second The spiders choose the(distinct) starting positions of all the bugs, with the requirement that the fly must begin at a vertex.Each bug knows the position of each other bug at all times, and the goal of the spiders is for at leastone of them to catch the fly What is the maximum c so that for any r < c, the fly can always avoidbeing caught?

29 How many positive integers a with a ≤ 154 are there such that the coefficient of xain the expansion of

(1 + x7+ x14+ · · · + x77)(1 + x11+ x22+ · · · + x77)

is zero?

30 The Lattice Point Jumping Frog jumps between lattice points in a coordinate plane that are exactly

1 unit apart The Lattice Point Jumping Frog starts at the origin and makes 8 jumps, ending atthe origin Additionally, it never lands on a point other than the origin more than once How manypossible paths could the frog have taken?

31 Let ABC be a triangle inscribed in circle Γ, centered at O with radius 333 Let M be the midpoint

of AB, N be the midpoint of AC, and D be the point where line AO intersects BC Given that lines

M N and BO concur on Γ and that BC = 665, find the length of segment AD

32 The sequence {an} satisfies a0= 201, a1= 2011, and an= 2an−1+ an−2for all n ≥ 2 Let

What is 1

S?

33 You are playing a game in which you have 3 envelopes, each containing a uniformly random amount

of money between 0 and 1000 dollars (That is, for any real 0 ≤ a < b ≤ 1000, the probability that theamount of money in a given envelope is between a and b is 1000b−a.) At any step, you take an envelope andlook at its contents You may choose either to keep the envelope, at which point you finish, or discard

it and repeat the process with one less envelope If you play to optimize your expected winnings, yourexpected winnings will be E What is bEc, the greatest integer less than or equal to E?

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34 Let p, q, r be real numbers satisfying

36 Let snbe the number of solutions to a1+ a2+ a3+ a4+ b1+ b2= n, where a1, a2, a3and a4are elements

of the set {2, 3, 5, 7} and b1 and b2 are elements of the set {1, 2, 3, 4} Find the number of n for which

sn is odd

37 In triangle ABC, AB = 1 and AC = 2 Suppose there exists a point P in the interior of triangleABC such that ∠P BC = 70◦, and that there are points E and D on segments AB and AC, such that

∠BP E = ∠EP A = 75◦ and ∠AP D = ∠DP C = 60◦ Let BD meet CE at Q, and let AQ meet BC

at F If M is the midpoint of BC, compute the degree measure of ∠M P F

38 Let S denote the sum of the 2011th powers of the roots of the polynomial (x − 20)(x − 21) · · · (x −

22010) − 1 How many 1’s are in the binary expansion of S?

39 For positive integers n, let ν3(n) denote the largest integer k such that 3k divides n Find the number

of subsets S (possibly containing 0 or 1 elements) of {1, 2, , 81} such that for any distinct a, b ∈ S,

Find the largest possible value of (xy + wz)2

42 In triangle ABC, sin ∠A = 45 and ∠A < 90◦ Let D be a point outside triangle ABC such that

∠BAD = ∠DAC and ∠BDC = 90◦ Suppose that AD = 1 and that BDCD = 32 If AB + AC can beexpressed in the form a

√ b

c where a, b, c are pairwise relatively prime integers, find a + b + c?

43 An integer x is selected at random between 1 and 2011! inclusive The probability that xx− 1 isdivisible by 2011 can be expressed in the form m

n, where m and n are relatively prime positive integers.Find m

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44 Given a set of points in space, a jump consists of taking two points in the set, P and Q, removing Pfrom the set, and replacing it with the reflection of P over Q Find the smallest number n such that forany set of n lattice points in 10-dimensional-space, it is possible to perform a finite number of jumps

so that some two points coincide

45 Let K1, K2, K3, K4, K5be 5 distinguishable keys, and let D1, D2, D3, D4, D5be 5 distinguishable doors.For 1 ≤ i ≤ 5, key Ki opens doors Diand Di+1(where D6= D1) and can only be used once The keysand doors are placed in some order along a hallway Key$ha walks into the hallway, picks a key andopens a door with it, such that she never obtains a key before all the doors in front of it are unlocked

In how many such ways can the keys and doors be ordered if Key$ha can open all the doors?

46 Let f is a function from the set of positive integers to itself such that f (x) ≤ x2for all natural numbers

x, and f (f (f (x))f (f (y))) = xy for all natural numbers x and y Find the number of possible values

of f (30)

47 Let ABCD be an isosceles trapezoid with bases AB = 5 and CD = 7 and legs BC = AD = 2√

10 Acircle ω with center O passes through A, B, C, and D Let M be the midpoint of segment CD, and ray

AM meet ω again at E Let N be the midpoint of BE and P be the intersection of BE with CD Let

Q be the intersection of ray ON with ray DC There is a point R on the circumcircle of P N Q suchthat ∠P RC = 45◦ The length of DR can be expressed in the form mn where m and n are relativelyprime positive integers What is m + n?

49 Find the magnitude of the product of all complex numbers c such that the recurrence defined by x1= 1,

x2= c2− 4c + 7, and xn+1= (c2− 2c)2xnxn−1+ 2xn− xn−1also satisfies x1006= 2011

50 In tetrahedron SABC, the circumcircles of faces SAB, SBC, and SCA each have radius 108 Theinscribed sphere of SABC, centered at I, has radius 35 Additionally, SI = 125 Let R is the largestpossible value of the circumradius of face ABC Give that R can be expressed in the formpm

n, where

m and n are relatively prime positive integers, find m + n

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Acknowledgments

Test Directors

Ray Li, Anderson Wang, and Alex Zhu

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October 18 - 29, 2013

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Contest Information

These rules supersede any rules found elsewhere about the OMO Please send any further questions directly

to the OMO Team at OnlineMathOpenTeam@gmail.com

Team Registration and Eligibility

Students may compete in teams of up to four people, but no student can belong to more than one team.Participants must not have graduated from high school (or the equivalent secondary school institution inother countries) Teams need not remain the same between the Fall and Spring contests, and students arepermitted to participate in one contest but not the other

Only one member on each team needs to register an account on the website Please check thewebsite, http://internetolympiad.org/pages/14-omo_info, for registration instructions

Note: when we say “up to four”, we really do mean “up to”! Because the time limit is so long, partial teamsare not significantly disadvantaged, and we welcome their participation

Contest Format and Rules

The 2013 Fall Contest will consist of 30 problems; the answer to each problem will be a nonnegativeinteger not exceeding 263− 2 = 9223372036854775806 The contest window will be October 18 - 29,

2013, from 7PM ET on the start day to 7PM ET on the end day There is no time limit other than thecontest window

1 Four-function calculators (calculators which can perform only the four basic arithmetic operations)are permitted on the Online Math Open Any other computational aids, including scientificcalculators, graphing calculators, or computer programs is prohibited All problems on theOnline Math Open are solvable without a calculator Four-function calculators are permitted only tohelp participants reduce computation errors

2 Drawing aids such as graph paper, ruler, and compass are permitted However, electronic drawingaids, such as Geogebra and graphing calculators, are not allowed Print and electronicpublications are also not allowed

3 Members of different teams cannot communicate with each other about the contest while the contest

is running

4 Your score is the number of questions answered correctly; that is, every problem is worth one point.Ties will be broken based on the ”hardest” problem that a team answered correctly Remaining tieswill be broken by the second hardest problem solved, and so on (Problem m is harder than problem

n if fewer teams solve problem m OR if the number of solves is equal and m > n.)

5 Participation in the Online Math Open is free

Clarifications and Results

Clarifications will be posted as they are answered For the most recent contests, they will be posted athttp://internetolympiad.org/pages/n/omo_problems If you have a question about problem wording,please email OnlineMathOpenTeam@gmail.com with “Clarification” in the subject We have the right to denyclarification requests that we feel we cannot answer

After the contest is over, we will release the answers to the problems within the next day Please do notdiscuss the test until answers are released If you have a protest about an answer, you may send an email

to OnlineMathOpenTeam@gmail.com (Include “Protest” in the subject) Results will be released in thefollowing weeks (Results will be counted only for teams that submit answers at least once Teams that onlyregister an account will not be listed in the final rankings.)

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OMO Fall 2013October 18 - 29, 2013

1 Determine the value of 142857 + 285714 + 428571 + 571428

2 The figure below consists of several unit squares, M of which are white and N of which are green.Compute 100M + N

3 A palindromic table is a 3 × 3 array of letters such that the words in each row and column read thesame forwards and backwards An example of such a table is shown below

5 A wishing well is located at the point (11, 11) in the xy-plane Rachelle randomly selects an integer

y from the set {0, 1, , 10} Then she randomly selects, with replacement, two integers a, b from theset {1, 2, , 10} The probability the line through (0, y) and (a, b) passes through the well can beexpressed as mn, where m and n are relatively prime positive integers Compute m + n

6 Find the number of integers n with n ≥ 2 such that the remainder when 2013 is divided by n is equal

to the remainder when n is divided by 3

7 Points M , N , P are selected on sides AB, AC, BC, respectively, of triangle ABC Find the area oftriangle M N P given that AM = M B = BP = 15 and AN = N C = CP = 25

8 Suppose that x1 < x2< · · · < xn is a sequence of positive integers such that xk divides xk+2 for each

k = 1, 2, , n − 2 Given that xn = 1000, what is the largest possible value of n?

9 Let AXY ZB be a regular pentagon with area 5 inscribed in a circle with center O Let Y0 denote thereflection of Y over AB and suppose C is the center of a circle passing through A, Y0 and B Computethe area of triangle ABC

10 In convex quadrilateral AEBC, ∠BEA = ∠CAE = 90◦ and AB = 15, BC = 14 and CA = 13 Let D

be the foot of the altitude from C to AB If ray CD meets AE at F , compute AE · AF

11 Four orange lights are located at the points (2, 0), (4, 0), (6, 0) and (8, 0) in the xy-plane Four yellowlights are located at the points (1, 0), (3, 0), (5, 0), (7, 0) Sparky chooses one or more of the lights toturn on In how many ways can he do this such that the collection of illuminated lights is symmetricaround some line parallel to the y-axis?

12 Let an denote the remainder when (n + 1)3 is divided by n3; in particular, a1 = 0 Compute theremainder when a1+ a2+ · · · + a2013is divided by 1000

1

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13 In the rectangular table shown below, the number 1 is written in the upper-left hand corner, and everynumber is the sum of the any numbers directly to its left and above The table extends infinitelydownwards and to the right

. . . . .Wanda the Worm, who is on a diet after a feast two years ago, wants to eat n numbers (not necessarilydistinct in value) from the table such that the sum of the numbers is less than one million However,she cannot eat two numbers in the same row or column (or both) What is the largest possible value

of n?

14 In the universe of Pi Zone, points are labeled with 2 × 2 arrays of positive reals One can teleport frompoint M to point M0 if M can be obtained from M0 by multiplying either a row or column by somepositive real For example, one can teleport from

to

1 20

3 40

and then to

1 20

6 80

A tourist attraction is a point where each of the entries of the associated array is either 1, 2, 4, 8 or

16 A company wishes to build a hotel on each of several points so that at least one hotel is accessiblefrom every tourist attraction by teleporting, possibly multiple times What is the minimum number

where f (n) denotes the nth positive integer which is not a perfect square

16 Al has the cards 1, 2, , 10 in a row in increasing order He first chooses the cards labeled 1, 2, and

3, and rearranges them among their positions in the row in one of six ways (he can leave the positionsunchanged) He then chooses the cards labeled 2, 3, and 4, and rearranges them among their positions

in the row in one of six ways (For example, his first move could have made the sequence 3, 2, 1, 4, 5, ,and his second move could have rearranged that to 2, 4, 1, 3, 5, ) He continues this process until hehas rearranged the cards with labels 8, 9, 10 Determine the number of possible orderings of cards hecan end up with

17 Let ABXC be a parallelogram Points K, P, Q lie on BC in this order such that BK = 13KC and

BP = P Q = QC = 13BC Rays XP and XQ meet AB and AC at D and E, respectively Supposethat AK ⊥ BC, EK − DK = 9 and BC = 60 Find AB + AC

18 Given an n × n grid of dots, let f (n) be the largest number of segments between adjacent dots whichcan be drawn such that (i) at most one segment is drawn between each pair of dots, and (ii) each dothas 1 or 3 segments coming from it (For example, f (4) = 16.) Compute f (2000)

19 Let σ(n) be the number of positive divisors of n, and let rad n be the product of the distinct primedivisors of n By convention, rad 1 = 1 Find the greatest integer not exceeding

20 A positive integer n is called mythical if every divisor of n is two less than a prime Find the uniquemythical number with the largest number of divisors

2

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21 Let ABC be a triangle with AB = 5, AC = 8, and BC = 7 Let D be on side AC such that AD = 5and CD = 3 Let I be the incenter of triangle ABC and E be the intersection of the perpendicularbisectors of ID and BC Suppose DE = a

√ b

c where a and c are relatively prime positive integers, and

b is a positive integer not divisible by the square of any prime Find a + b + c

22 Find the sum of all integers m with 1 ≤ m ≤ 300 such that for any integer n with n ≥ 2, if 2013mdivides nn− 1 then 2013m also divides n − 1

23 Let ABCDE be a regular pentagon, and let F be a point on AB with ∠CDF = 55◦ Suppose F Cand BE meet at G, and select H on the extension of CE past E such that ∠DHE = ∠F DG Findthe measure of ∠GHD, in degrees

24 The real numbers a0, a1, , a2013 and b0, b1, , b2013 satisfy an = 631√

25 Let ABCD be a quadrilateral with AD = 20 and BC = 13 The area of 4ABC is 338 and the area

of 4DBC is 212 Compute the smallest possible perimeter of ABCD

26 Let ABC be a triangle with AB = 13, AC = 25, and tan A = 3

4 Denote the reflections of B, C across

AC, AB by D, E, respectively, and let O be the circumcenter of triangle ABC Let P be a point suchthat 4DP O ∼ 4P EO, and let X and Y be the midpoints of the major and minor arcs dBC of thecircumcircle of triangle ABC Find P X · P Y

27 Ben has a big blackboard, initially empty, and Francisco has a fair coin Francisco flips the coin 2013times On the nth flip (where n = 1, 2, , 2013), Ben does the following if the coin flips heads:(i) If the blackboard is empty, Ben writes n on the blackboard

(ii) If the blackboard is not empty, let m denote the largest number on the blackboard If m2+ 2n2

is divisible by 3, Ben erases m from the blackboard; otherwise, he writes the number n

No action is taken when the coin flips tails If probability that the blackboard is empty after all 2013flips is 2k2u+1(2v+1), where u, v, and k are nonnegative integers, compute k

28 Let n denote the product of the first 2013 primes Find the sum of all primes p with 20 ≤ p ≤ 150such that

(i) p+12 is even but is not a power of 2, and

(ii) there exist pairwise distinct positive integers a, b, c for which

an(a − b)(a − c) + bn(b − c)(b − a) + cn(c − a)(c − b)

is divisible by p but not p2

29 Kevin has 255 cookies, each labeled with a unique nonempty subset of {1, 2, 3, 4, 5, 6, 7, 8} Each day,

he chooses one cookie uniformly at random out of the cookies not yet eaten Then, he eats that cookie,and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses thecookie labeled with {1, 2}, he eats that cookie as well as the cookies with {1} and {2}) The expectedvalue of the number of days that Kevin eats a cookie before all cookies are gone can be expressed inthe form mn, where m and n are relatively prime positive integers Find m + n

30 Let P (t) = t3+ 27t2 + 199t + 432 Suppose a, b, c, and x are distinct positive reals such that

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The Online Math Open Winter Contest January 4, 2013–January 14, 2013

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Contest Directors

Ray Li, James Tao, Victor Wang

Evan Chen, Ray Li, Victor Wang

Additional Problem Contributors

James Tao, Anderson Wang, David Yang, Alex Zhu

Proofreaders and Test Solvers

Evan Chen, Calvin Deng, Mitchell Lee, James Tao, Anderson Wang, David Yang, Alex Zhu

Website Manager

Ray Li

LATEX/Document Manager

Evan Chen

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Contest Information

Format

The test will start Friday, January 4 and end Monday, January 14 You will have until 7pm EST on January

14 to submit your answers The test consists of 50 short answer questions, each of which has a nonnegativeinteger answer The problem difficulties range from those of AMC problems to those of Olympiad problems.Problems are ordered in roughly increasing order of difficulty

Team Guidelines

Students may compete in teams of up to four people Participating students must not have graduated fromhigh school International students may participate No student can be a part of more than one team Themembers of each team do not get individual accounts; they will all share the team account

Each team will submit its final answers through its team account Though teams can save drafts for theiranswers, the current interface does not allow for much flexibility in communication between team members

Aids

Drawing aids such as graph paper, ruler, and compass are permitted However, electronic drawing aids arenot allowed This is includes (but is not limited to) Geogebra and graphing calculators Published printand electronic resources are not permitted (This is a change from last year’s rules.)

Four-function calculators are permitted on the Online Math Open That is, calculators which perform onlythe four basic arithmetic operations (+-*/) may be used Any other computational aids such as scientificand graphing calculators, computer programs and applications such as Mathematica, and online databasesare prohibited All problems on the Online Math Open are solvable without a calculator Four-functioncalculators are permitted only to help participants reduce computation errors

Clarifications

Clarifications will be posted as they are answered For the Fall 2012-2013 Contest, they will be posted at here

If you have a question about a problem, please email OnlineMathOpenTeam@gmail.com with “Clarification”

in the subject We have the right to deny clarification requests that we feel we cannot answer

Scoring

Each problem will be worth one point Ties will be broken based on the “hardest” problem that a teamanswered correctly Remaining ties will be broken by the second hardest problem solved, and so on Problem

X is defined to be “harder” than Problem Y if and only if

(i) X was solved by less teams than Y , OR

(ii) X and Y were solved by the same number of teams and X appeared later in the test than Y

Note: This is a change from prior tiebreaking systems However, we will still order theproblems by approximate difficulty

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January 2012 Winter OMO 2012-2013 Page 1

1 Let x be the answer to this problem For what real number a is the answer to this problem also a−x?

2 The number 123454321 is written on a blackboard Evan walks by and erases some (but not all) of thedigits, and notices that the resulting number (when spaces are removed) is divisible by 9 What is thefewest number of digits he could have erased?

3 Three lines m, n, and ` lie in a plane such that no two are parallel Lines m and n meet at an acuteangle of 14◦, and lines m and ` meet at an acute angle of 20◦ Find, in degrees, the sum of all possibleacute angles formed by lines n and `

4 For how many ordered pairs of positive integers (a, b) with a, b < 1000 is it true that a times b is equal

to b2 divided by a? For example, 3 times 9 is equal to 92divided by 3

Figure 1: xkcd 759

5 At the Mountain School, Micchell is assigned a submissiveness rating of 3.0 or 4.0 for each class hetakes His college potential is then defined as the average of his submissiveness ratings over all classestaken After taking 40 classes, Micchell has a college potential of 3.975 Unfortunately, he needs acollege potential of at least 3.995 to get into the South Harmon Institute of Technology Otherwise, hebecomes a rock Assuming he receives a submissiveness rating of 4.0 in every class he takes from now

on, how many more classes does he need to take in order to get into the South Harmon Institute ofTechnology?

6 Circle S1 has radius 5 Circle S2 has radius 7 and has its center lying on S1 Circle S3has an integerradius and has its center lying on S2 If the center of S1lies on S3, how many possible values are therefor the radius of S3?

7 Jacob’s analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have beenerased, so he doesn’t know which tick mark corresponds to which hour Jacob takes an arbitrary tickmark and measures clockwise to the hour hand and minute hand He measures that the minute hand

is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the sametick mark If it is currently morning, how many minutes past midnight is it?

8 How many ways are there to choose (not necessarily distinct) integers a, b, c from the set {1, 2, 3, 4}such that a(bc) is divisible by 4?

9 David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones An operation consists ofremoving three objects, no two of the same type What is the maximum number of operations he canpossibly perform?

10 At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and

5 oranges costs 13 dollars Given that apples and oranges can only be bought in these two packages,what is the minimum nonzero amount of dollars that must be spent to have an equal number of applesand oranges?

11 Let A, B, and C be distinct points on a line with AB = AC = 1 Square ABDE and equilateraltriangle ACF are drawn on the same side of line BC What is the degree measure of the acute angleformed by lines EC and BF ?

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12 There are 25 ants on a number line; five at each of the coordinates 1, 2, 3, 4, and 5 Each minute,one ant moves from its current position to a position one unit away What is the minimum number ofminutes that must pass before it is possible for no two ants to be on the same coordinate?

13 There are three flies of negligible size that start at the same position on a circular track with ference 1000 meters They fly clockwise at speeds of 2, 6, and k meters per second, respectively, where

circum-k is some positive integer with 7 ≤ circum-k ≤ 2013 Suppose that at some point in time, all three flies meet

at a location different from their starting point How many possible values of k are there?

14 What is the smallest perfect square larger than 1 with a perfect square number of positive integerfactors?

15 A permutation a1, a2, , a13 of the numbers from 1 to 13 is given such that ai> 5 for i = 1, 2, 3, 4, 5.Determine the maximum possible value of

aa1+ aa2+ aa3+ aa4+ aa5

16 Let S1 and S2 be two circles intersecting at points A and B Let C and D be points on S1 and

S2 respectively such that line CD is tangent to both circles and A is closer to line CD than B If

∠BCA = 52◦ and ∠BDA = 32◦, determine the degree measure of ∠CBD

17 Determine the number of ordered pairs of positive integers (x, y) with y < x ≤ 100 such that x2− y2

and x3− y3 are relatively prime (Two numbers are relatively prime if they have no common factorother than 1.)

18 Determine the absolute value of the sum

b2013 sin 0◦c + b2013 sin 1◦c + · · · + b2013 sin 359◦c,where bxc denotes the greatest integer less than or equal to x

(You may use the fact that sin n◦ is irrational for positive integers n not divisible by 30.)

19 A, B, C are points in the plane such that ∠ABC = 90◦ Circles with diameters BA and BC meet at

D If BA = 20 and BC = 21, then the length of segment BD can be expressed in the form mn where

m and n are relatively prime positive integers What is m + n?

20 Let a1, a2, , a2013 be a permutation of the numbers from 1 to 2013 Let An = a1 +a2+···+an

n = 1, 2, , 2013 If the smallest possible difference between the largest and smallest values of

A1, A2, , A2013is mn, where m and n are relatively prime positive integers, find m + n

21 Dirock has a very neat rectangular backyard that can be represented as a 32 × 32 grid of unit squares.The rows and columns are each numbered 1, 2, , 32 Dirock is very fond of rocks, and places a rock

in every grid square whose row and column number are both divisible by 3 Dirock would like to build

a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of theyard such that

a) The fence does not pass through any grid squares containing rocks;

b) The interior of the fence contains exactly 5 rocks

In how many ways can this be done?

22 In triangle ABC, AB = 28, AC = 36, and BC = 32 Let D be the point on segment BC satisfying

∠BAD = ∠DAC, and let E be the unique point such that DE k AB and line AE is tangent to thecircumcircle of ABC Find the length of segment AE

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23 A set of 10 distinct integers S is chosen Let M be the number of nonempty subsets of S whoseelements have an even sum What is the minimum possible value of M ?

Clarifications

• S is the “set of 10 distinct integers” from the first sentence

24 For a permutation π of the integers from 1 to 10, define

S(π) =

9

X

i=1

(π(i) − π(i + 1)) · (4 + π(i) + π(i + 1)),

where π(i) denotes the ith element of the permutation Suppose that M is the maximum possible value

of S(π) over all permutations π of the integers from 1 to 10 Determine the number of permutations

π for which S(π) = M

25 Positive integers x, y, z ≤ 100 satisfy

1099x + 901y + 1110z = 59800109x + 991y + 101z = 44556Compute 10000x + 100y + z

26 In triangle ABC, F is on segment AB such that CF bisects ∠ACB Points D and E are on line CFsuch that lines AD, BE are perpendicular to CF M is the midpoint of AB If M E = 13, AD = 15,and BE = 25, find AC + CB

27 Geodude wants to assign one of the integers 1, 2, 3, , 11 to each lattice point (x, y, z) in a 3D Cartesiancoordinate system In how many ways can Geodude do this if for every lattice parallelogram ABCD,the positive difference between the sum of the numbers assigned to A and C and the sum of thenumbers assigned to B and D must be a multiple of 11? (A lattice point is a point with all integercoordinates A lattice parallelogram is a parallelogram with all four vertices lying on lattice points.)Clarifications

• The “positive difference” between two real numbers x and y is the quantity |x − y|

28 Let S be the set of all lattice points (x, y) in the plane satisfying |x| + |y| ≤ 10 Let P1, P2, , P2013

be a sequence of 2013 (not necessarily distinct) points such that for every point Q in S, there exists

at least one index i such that 1 ≤ i ≤ 2013 and Pi = Q Suppose that the minimum possible value

of |P1P2| + |P2P3| + · · · + |P2012P2013| can be expressed in the form a + b√c, where a, b, c are positiveintegers and c is not divisible by the square of any prime Find a + b + c (A lattice point is a pointwith all integer coordinates.)

30 Pairwise distinct points P1, P2, , P16lie on the perimeter of a square with side length 4 centered at

O such that |PiPi+1| = 1 for i = 1, 2, , 16 (We take P17 to be the point P1.) We construct points

Q1, Q2, , Q16 as follows: for each i, a fair coin is flipped If it lands heads, we define Qi to be Pi;otherwise, we define Qi to be the reflection of Pi over O (So, it is possible for some of the Qi tocoincide.) Let D be the length of the vector−−→

OQ1+−−→

OQ2+ · · · +−−−→

OQ16 Compute the expected value

of D2

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31 Beyond the Point of No Return is a large lake containing 2013 islands arranged at the vertices of aregular 2013-gon Adjacent islands are joined with exactly two bridges Christine starts on one of theislands with the intention of burning all the bridges Each minute, if the island she is on has at leastone bridge still joined to it, she randomly selects one such bridge, crosses it, and immediately burns it.Otherwise, she stops

If the probability Christine burns all the bridges before she stops can be written as m

n for relativelyprime positive integers m and n, find the remainder when m + n is divided by 1000

32 In 4ABC with incenter I, AB = 61, AC = 51, and BC = 71 The circumcircles of triangles AIBand AIC meet line BC at points D (D 6= B) and E (E 6= C), respectively Determine the length ofsegment DE

33 Let n be a positive integer E Chen and E Chen play a game on the n2 points of an n × n latticegrid They alternately mark points on the grid such that no player marks a point that is on or inside

a non-degenerate triangle formed by three marked points Each point can be marked only once Thegame ends when no player can make a move, and the last player to make a move wins Determine thenumber of values of n between 1 and 2013 (inclusive) for which the first player can guarantee a win,regardless of the moves that the second player makes

34 For positive integers n, let s(n) denote the sum of the squares of the positive integers less than or equal

to n that are relatively prime to n Find the greatest integer less than or equal to

X

n|2013

s(n)

n2 ,where the summation runs over all positive integers n dividing 2013

35 The rows and columns of a 7 × 7 grid are each numbered 1, 2, , 7 In how many ways can one choose

8 cells of this grid such that for every two chosen cells X and Y , either the positive difference of theirrow numbers is at least 3, or the positive difference of their column numbers is at least 3?

Clarifications

• The “or” here is inclusive (as by convention, despite the “either”), i.e X and Y are permitted ifand only if they satisfy the row condition, the column condition, or both

36 Let ABCD be a nondegenerate isosceles trapezoid with integer side lengths such that BC k AD and

AB = BC = CD Given that the distance between the incenters of triangles ABD and ACD is 8!,determine the number of possible lengths of segment AD

37 Let M be a positive integer At a party with 120 people, 30 wear red hats, 40 wear blue hats, and

50 wear green hats Before the party begins, M pairs of people are friends (Friendship is mutual.)Suppose also that no two friends wear the same colored hat to the party

During the party, X and Y can become friends if and only if the following two conditions hold:a) There exists a person Z such that X and Y are both friends with Z (The friendship(s) between

Z, X and Z, Y could have been formed during the party.)

b) X and Y are not wearing the same colored hat

Suppose the party lasts long enough so that all possible friendships are formed Let M1 be the largestvalue of M such that regardless of which M pairs of people are friends before the party, there willalways be at least one pair of people X and Y with different colored hats who are not friends after theparty Let M2 be the smallest value of M such that regardless of which M pairs of people are friendsbefore the party, every pair of people X and Y with different colored hats are friends after the party.Find M + M

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Clarifications

• The definition of M2 should read, “Let M2be the smallest value of M such that ” An earlierversion of the test read “largest value of M ”

38 Triangle ABC has sides AB = 25, BC = 30, and CA = 20 Let P, Q be the points on segments

AB, AC, respectively, such that AP = 5 and AQ = 4 Suppose lines BQ and CP intersect at R andthe circumcircles of 4BP R and 4CQR intersect at a second point S 6= R If the length of segment

SA can be expressed in the form √ m

n for positive integers m, n, where n is not divisible by the square

of any prime, find m + n

39 Find the number of 8-digit base-6 positive integers (a1a2a3a4a5a6a7a8)6(with leading zeros permitted)such that (a1a2 a8)6| (ai+1ai+2 ai+8)6 for i = 1, 2, , 7, where indices are taken modulo 8 (so

a9= a1, a10= a2, and so on)

40 Let ABC be a triangle with AB = 13, BC = 14, and AC = 15 Let M be the midpoint of BC andlet Γ be the circle passing through A and tangent to line BC at M Let Γ intersect lines AB and AC

at points D and E, respectively, and let N be the midpoint of DE Suppose line M N intersects lines

AB and AC at points P and O, respectively If the ratio M N : N O : OP can be written in the form

a : b : c with a, b, c positive integers satisfying gcd(a, b, c) = 1, find a + b + c

41 While there do not exist pairwise distinct real numbers a, b, c satisfying a2+ b2+ c2= ab + bc + ca, there

do exist complex numbers with that property Let a, b, c be complex numbers such that a2+ b2+ c2=

ab + bc + ca and |a + b + c| = 21 Given that |a − b| = 2√

3, |a| = 3√

3, compute |b|2+ |c|2.Clarifications

• The problem should read |a + b + c| = 21 An earlier version of the test read |a + b + c| = 7; thatvalue is incorrect

• |b|2+ |c|2 should be a positive integer, not a fraction; an earlier version of the test read “ forrelatively prime positive integers m and n Find m + n.”

44 Suppose tetrahedron P ABC has volume 420 and satisfies AB = 13, BC = 14, and CA = 15 Theminimum possible surface area of P ABC can be written as m+n√

k, where m, n, k are positive integersand k is not divisible by the square of any prime Compute m + n + k

45 Let N denote the number of ordered 2011-tuples of positive integers (a1, a2, , a2011) with 1 ≤

a1, a2, , a2011≤ 20112 such that there exists a polynomial f of degree 4019 satisfying the followingthree properties:

• f (n) is an integer for every integer n;

• 20112| f (i) − aifor i = 1, 2, , 2011;

• 20112| f (n + 2011) − f (n) for every integer n

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Find the remainder when N is divided by 1000

46 Let ABC be a triangle with ∠B − ∠C = 30◦ Let D be the point where the A-excircle touches line

BC, O the circumcenter of triangle ABC, and X, Y the intersections of the altitude from A with theincircle with X in between A and Y Suppose points A, O and D are collinear If the ratio AXAO can beexpressed in the form a+b

√ c

d for positive integers a, b, c, d with gcd(a, b, d) = 1 and c not divisible bythe square of any prime, find a + b + c + d

47 Let f (x, y) be a function from ordered pairs of positive integers to real numbers such that

f (1, x) = f (x, 1) = 1

x and f (x + 1, y + 1)f (x, y) − f (x, y + 1)f (x + 1, y) = 1for all ordered pairs of positive integers (x, y) If f (100, 100) = mn for two relatively prime positiveintegers m, n, compute m + n

48 ω is a complex number such that ω2013= 1 and ωm6= 1 for m = 1, 2, , 2012 Find the number ofordered pairs of integers (a, b) with 1 ≤ a, b ≤ 2013 such that

2, CB = 6720, and ∠C = 45◦ Let K, L, M lie on BC, CA, and AB such that

AK ⊥ BC, BL ⊥ CA, and AM = BM Let N , O, P lie on KL, BA, and BL such that AN = KN ,

BO = CO, and A lies on line N P If H is the orthocenter of 4M OP , compute HK2

Clarifications

• Without further qualification, “XY” denotes line XY

50 Let S denote the set of words W = w1w2 wn of any length n ≥ 0 (including the empty stringλ), with each letter wi from the set {x, y, z} Call two words U, V similar if we can insert a string

s ∈ {xyz, yzx, zxy} of three consecutive letters somewhere in U (possibly at one of the ends) to obtain

V or somewhere in V (again, possibly at one of the ends) to obtain U , and say a word W is trivial

if for some nonnegative integer m, there exists a sequence W0, W1, , Wm such that W0 = λ is theempty string, Wm= W , and Wi, Wi+1 are similar for i = 0, 1, , m − 1 Given that for two relativelyprime positive integers p, q we have

p

q =X

n≥0

f (n) 2258192

n

,

where f (n) denotes the number of trivial words in S of length 3n (in particular, f (0) = 1), find p+q

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The Online Math Open Spring Contest

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Contest Information

Students may compete in teams of up to four people, but no student can belong to more than one team.Participants must not have graduated from high school (or the equivalent secondary school institution inother countries) Teams need not remain the same between the Fall and Spring contests, and students arepermitted to participate in whichever contests they like

The 2014 Spring Contest will consist of 30 problems; the answer to each problem will be a nonnegativeinteger not exceeding 232− 1 = 4294967295 The contest window will be April 4 - 15, 2014, from 7PM

ET on the start day to 7PM ET on the end day There is no time limit other than the contest window

1 Four-function calculators (calculators which can perform only the four basic arithmetic operations) arepermitted on the Online Math Open Any other computational aids are prohibited, including scientificcalculators, graphing calculators, or computer programs All problems on the Online Math Open aresolvable without a calculator

2 Drawing aids such as graph paper, ruler, and compass are permitted However, electronic drawingaids, such as Geogebra and graphing calculators, are not allowed Print and electronic publications arealso not allowed

3 Members of different teams cannot communicate about the contest until the contest ends

4 Your score is the number of questions answered correctly; that is, every problem is worth one point.Ties will be broken based on the “hardest” problem that a team answered correctly Problem m isharder than problem n if fewer teams solve problem m, or if the number of solves is equal and m > n.Remaining ties will be broken by the second hardest problem solved, and so on

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OMO Spring 2014April 4 - 15, 2014

1 In English class, you have discovered a mysterious phenomenon – if you spend n hours on an essay,your score on the essay will be 100 (1 − 4−n) points if 2n is an integer, and 0 otherwise For example,

if you spend 30 minutes on an essay you will get a score of 50, but if you spend 35 minutes on theessay you somehow do not earn any points

It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start

of class If you can only work on one essay at a time, what is the maximum possible average of youressay scores?

2 Consider two circles of radius one, and let O and O0denote their centers Point M is selected on eithercircle If OO0 = 2014, what is the largest possible area of triangle OM O0?

3 Suppose that m and n are relatively prime positive integers with A = mn, where

A = 2 + 4 + 6 + · · · + 2014

1 + 3 + 5 + · · · + 2013−1 + 3 + 5 + · · · + 2013

2 + 4 + 6 + · · · + 2014.Find m In other words, find the numerator of A when A is written as a fraction in simplest form

4 The integers 1, 2, , n are written in order on a long slip of paper The slip is then cut into fivepieces, so that each piece consists of some (nonempty) consecutive set of integers The averages of thenumbers on the five slips are 1234, 345, 128, 19, and 9.5 in some order Compute n

5 Joe the teacher is bad at rounding Because of this, he has come up with his own way to round grades,where a grade is a nonnegative decimal number with finitely many digits after the decimal point.Given a grade with digits a1a2 am.b1b2 bn, Joe first rounds the number to the nearest 10−n+1thplace He then repeats the procedure on the new number, rounding to the nearest 10−n+2th, thenrounding the result to the nearest 10−n+3th, and so on, until he obtains an integer For example, herounds the number 2014.456 via 2014.456 → 2014.46 → 2014.5 → 2015

There exists a rational number M such that a grade x gets rounded to at least 90 if and only if x ≥ M

If M = pq for relatively prime integers p and q, compute p + q

6 Let Ln be the least common multiple of the integers 1, 2, , n For example, L10 = 2,520 and

L30= 2,329,089,562,800 Find the remainder when L31 is divided by 100,000

7 How many integers n with 10 ≤ n ≤ 500 have the property that the hundreds digit of 17n and 17n + 17are different?

8 Let a1, a2, a3, a4, a5 be real numbers satisfying

2a1+ a2+ a3+ a4+ a5= 1 +18a4

2a2+ a3+ a4+ a5= 2 +14a3

2a3+ a4+ a5= 4 +12a2

2a4+ a5= 6 + a1Compute a1+ a2+ a3+ a4+ a5

9 Eighteen students participate in a team selection test with three problems, each worth up to sevenpoints All scores are nonnegative integers After the competition, the results are posted by Evan

in a table with 3 columns: the student’s name, score, and rank (allowing ties), respectively Here, astudent’s rank is one greater than the number of students with strictly higher scores (for example, ifseven students score 0, 0, 7, 8, 8, 14, 21 then their ranks would be 6, 6, 5, 3, 3, 2, 1 respectively)

When Richard comes by to read the results, he accidentally reads the rank column as the scorecolumn and vice versa Coincidentally, the results still made sense! If the scores of the studentswere x1≤ x2≤ · · · ≤ x18, determine the number of possible values of the 18-tuple (x1, x2, , x18) Inother words, determine the number of possible multisets (sets with repetition) of scores

1

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10 Let A1A2 A4000 be a regular 4000-gon Let X be the foot of the altitude from A1986 onto diagonal

A1000A3000, and let Y be the foot of the altitude from A2014 onto A2000A4000 If XY = 1, what is thearea of square A500A1500A2500A3500?

11 Let X be a point inside convex quadrilateral ABCD with ∠AXB + ∠CXD = 180◦ If AX = 14,

BX = 11, CX = 5, DX = 10, and AB = CD, find the sum of the areas of 4AXB and 4CXD

12 The points A, B, C, D, E lie on a line ` in this order Suppose T is a point not on ` such that

∠BT C = ∠DT E, and AT is tangent to the circumcircle of triangle BT E If AB = 2, BC = 36, and

holds for all nonzero real numbers x Find g(2)

14 Let ABC be a triangle with incenter I and AB = 1400, AC = 1800, BC = 2014 The circle centered

at I passing through A intersects line BC at two points X and Y Compute the length XY

15 In Prime Land, there are seven major cities, labelled C0, C1, , C6 For convenience, we let Cn+7= Cn

for each n = 0, 1, , 6; i.e we take the indices modulo 7 Al initially starts at city C0

Each minute for ten minutes, Al flips a fair coin If the coin land heads, and he is at city Ck, he moves

to city C2k; otherwise he moves to city C2k+1 If the probability that Al is back at city C0 after 10moves is 1024m , find m

16 Say a positive integer n is radioactive if one of its prime factors is strictly greater than √

n Forexample, 2012 = 22· 503, 2013 = 3 · 11 · 61 and 2014 = 2 · 19 · 53 are all radioactive, but 2015 = 5 · 13 · 31

is not How many radioactive numbers have all prime factors less than 30?

17 Let AXY BZ be a convex pentagon inscribed in a circle with diameter AB The tangent to the circle

at Y intersects lines BX and BZ at L and K, respectively Suppose that AY bisects ∠LAZ and

AY = Y Z If the minimum possible value of

AK

ALAB

20 Let ABC be an acute triangle with circumcenter O, and select E on AC and F on AB so that

BE ⊥ AC, CF ⊥ AB Suppose ∠EOF − ∠A = 90◦and ∠AOB − ∠B = 30◦ If the maximum possiblemeasure of ∠C is mn · 180◦ for some positive integers m and n with m < n and gcd(m, n) = 1, compute

m + n

21 Let b = 12(−1 + 3√

5) Determine the number of rational numbers which can be written in the form

a2014b2014+ a2013b2013+ · · · + a1b + a0where a0, a1, , a2014are nonnegative integers less than b

2

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22 Let f (x) be a polynomial with integer coefficients such that f (15)f (21)f (35) − 10 is divisible by 105.Given f (−34) = 2014 and f (0) ≥ 0, find the smallest possible value of f (0)

23 Let Γ1and Γ2be circles in the plane with centers O1and O2and radii 13 and 10, respectively Assume

O1O2= 2 Fix a circle Ω with radius 2, internally tangent to Γ1 at P and externally tangent to Γ2 at

Q Let ω be a second variable circle internally tangent to Γ1 at X and externally tangent to Γ2at Y Line P Q meets Γ2 again at R, line XY meets Γ2 again at Z, and lines P Z and XR meet at M

As ω varies, the locus of point M encloses a region of area pqπ, where p and q are relatively primepositive integers Compute p + q

24 Let P denote the set of planes in three-dimensional space with positive x, y, and z intercepts summing

to one A point (x, y, z) with min{x, y, z} > 0 lies on exactly one plane in P What is the maximumpossible integer value of 14x2+ 2y2+ 16z2−1

qfor relatively prime positive integers p, q, find p + q

26 Qing initially writes the ordered pair (1, 0) on a blackboard Each minute, if the pair (a, b) is on theboard, she erases it and replaces it with one of the pairs (2a − b, a), (2a + b + 2, a) or (a + 2b + 2, b).Eventually, the board reads (2014, k) for some nonnegative integer k How many possible values of kare there?

27 A frog starts at 0 on a number line and plays a game On each turn the frog chooses at random tojump 1 or 2 integers to the right or left It stops moving if it lands on a nonpositive number or anumber on which it has already landed If the expected number of times it will jump is pq for relativelyprime positive integers p and q, find p + q

28 In the game of Nim, players are given several piles of stones On each turn, a player picks a nonemptypile and removes any positive integer number of stones from that pile The player who removes thelast stone wins, while the first player who cannot move loses

Alice, Bob, and Chebyshev play a 3-player version of Nim where each player wants to win but avoidslosing at all costs (there is always a player who neither wins nor loses) Initially, the piles have sizes

43, 99, x, y, where x and y are positive integers Assuming that the first player loses when all playersplay optimally, compute the maximum possible value of xy

29 Let ABCD be a tetrahedron whose six side lengths are all integers, and let N denote the sum ofthese side lengths There exists a point P inside ABCD such that the feet from P onto the faces

of the tetrahedron are the orthocenter of 4ABC, centroid of 4BCD, circumcenter of 4CDA, andorthocenter of 4DAB If CD = 3 and N < 100,000, determine the maximum possible value of N

30 For a positive integer n, an n-branch B is an ordered tuple (S1, S2, , Sm) of nonempty sets (where

m is any positive integer) satisfying S1⊂ S2⊂ · · · ⊂ Sm⊆ {1, 2, , n} An integer x is said to appear

in B if it is an element of the last set Sm Define an n-plant to be an (unordered) set of n-branches{B1, B2, , Bk}, and call it perfect if each of 1, 2, , n appears in exactly one of its branches.Let Tn be the number of distinct perfect n-plants (where T0= 1), and suppose that for some positivereal number x we have the convergence

ln



X

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October 17 - 28, 2014

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Contest Information

Students may compete in teams of up to four people, but no student can belong to more than one team.Participants must not have graduated from high school (or the equivalent secondary school institution inother countries) Teams need not remain the same between the Fall and Spring contests, and students arepermitted to participate in whichever contests they like

The 2014 Fall Contest will consist of 30 problems; the answer to each problem will be an integer between 0and 231− 1 = 2147483647 inclusive The contest window will be

October 17 - 28, 2014from 7PM ET on the start day to 7PM ET on the end day There is no time limit other than the contestwindow

1 Four-function calculators (calculators which can perform only the four basic arithmetic operations)are permitted on the Online Math Open Any other computational aids, including scientificcalculators, graphing calculators, or computer programs is prohibited All problems on theOnline Math Open are solvable without a calculator Four-function calculators are permitted only tohelp participants reduce computation errors

2 Drawing aids such as graph paper, ruler, and compass are permitted However, electronic drawingaids, such as Geogebra and graphing calculators, are not allowed Print and electronic publications arealso not allowed

3 Members of different teams cannot communicate with each other about the contest while the contest

is running

4 Your score is the number of questions answered correctly; that is, every problem is worth one point.Ties will be broken based on the “hardest” problem that a team answered correctly Remaining tieswill be broken by the second hardest problem solved, and so on (Problem m is harder than problem

n if fewer teams solve problem m OR if the number of solves is equal and m > n.)

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1 Carl has a rectangle whose side lengths are positive integers This rectangle has the property thatwhen he increases the width by 1 unit and decreases the length by 1 unit, the area increases by xsquare units What is the smallest possible positive value of x?

2 Suppose (an), (bn), (cn) are arithmetic progressions Given that a1+ b1+ c1= 0 and a2+ b2+ c2= 1,compute a2014+ b2014+ c2014

3 Let B = (20, 14) and C = (18, 0) be two points in the plane For every line ` passing through B, wecolor red the foot of the perpendicular from C to ` The set of red points enclose a bounded region ofarea A Find bAc (that is, find the greatest integer not exceeding A)

4 A crazy physicist has discovered a new particle called an emon He starts with two emons in the plane,situated a distance 1 from each other He also has a crazy machine which can take any two emons andcreate a third one in the plane such that the three emons lie at the vertices of an equilateral triangle.After he has five total emons, let P be the product of the 52 = 10 distances between the 10 pairs ofemons Find the greatest possible value of P2

5 A crazy physicist has discovered a new particle called an omon He has a machine, which takes twoomons of mass a and b and entangles them; this process destroys the omon with mass a, preserves theone with mass b, and creates a new omon whose mass is 1

2(a + b) The physicist can then repeat theprocess with the two resulting omons, choosing which omon to destroy at every step The physicistinitially has two omons whose masses are distinct positive integers less than 1000 What is the maxi-mum possible number of times he can use his machine without producing an omon whose mass is not

an integer?

6 For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure

a multiple of 10◦ She doesn’t want her triangle to have any special properties, so none of the anglescan measure 30◦ or 60◦, and the triangle should definitely not be isosceles

How many different triangles can Tina draw? (Similar triangles are considered the same.)

7 Define the function f (x, y, z) by

f (x, y, z) = xyz− xzy + yzx− yxz+ zxy.Evaluate f (1, 2, 3) + f (1, 3, 2) + f (2, 1, 3) + f (2, 3, 1) + f (3, 1, 2) + f (3, 2, 1)

8 Let a and b be randomly selected three-digit integers and suppose a > b We say that a is clearly biggerthan b if each digit of a is larger than the corresponding digit of b If the probability that a is clearlybigger than b is mn, where m and n are relatively prime integers, compute m + n

9 Let N = 2014!+2015!+2016!+· · ·+9999! How many zeros are at the end of the decimal representation

11 Given a triangle ABC, consider the semicircle with diameter EF on BC tangent to AB and AC If

BE = 1, EF = 24, and F C = 3, find the perimeter of 4ABC

12 Let a, b, c be positive real numbers for which

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13 Two ducks, Wat and Q, are taking a math test with 1022 other ducklings The test has 30 questions,and the nth question is worth n points The ducks work independently on the test Wat gets thenth problem correct with probability n12 while Q gets the nth problem correct with probability n+11 Unfortunately, the remaining ducklings each answer all 30 questions incorrectly

Just before turning in their test, the ducks and ducklings decide to share answers! On any questionwhich Wat and Q have the same answer, the ducklings change their answers to agree with them Afterthis process, what is the expected value of the sum of all 1024 scores?

14 What is the greatest common factor of 12345678987654321 and 12345654321?

15 Let φ = 1+

√ 5

2 A base-φ number (anan−1 a1a0)φ, where 0 ≤ an, an−1, , a0 ≤ 1 are integers, isdefined by

(anan−1 a1a0)φ= an· φn+ an−1· φn−1+ + a1· φ1+ a0.Compute the number of base-φ numbers (bjbj−1 b1b0)φ which satisfy bj6= 0 and

17 Let ABC be a triangle with area 5 and BC = 10 Let E and F be the midpoints of sides AC and ABrespectively, and let BE and CF intersect at G Suppose that quadrilateral AEGF can be inscribed

in a circle Determine the value of AB2+ AC2

18 We select a real number α uniformly and at random from the interval (0, 500) Define

S = 1α

Let p denote the probability that S ≥ 1200 Compute 1000p

19 In triangle ABC, AB = 3, AC = 5, and BC = 7 Let E be the reflection of A over BC, and letline BE meet the circumcircle of ABC again at D Let I be the incenter of 4ABD Given thatcos2

∠AEI = mn, where m and n are relatively prime positive integers, determine m + n

20 Let n = 2188 = 37+ 1 and let A(0)0 , A(0)1 , , A(0)n−1 be the vertices of a regular n-gon (in that order)with center O For i = 1, 2, , 7 and j = 0, 1, , n − 1, let A(i)j denote the centroid of the triangle

4A(i−1)j A(i−1)j+37−iA(i−1)j+2·37−i.Here the subscripts are taken modulo n If

|OA(7)2014|

|OA(0)2014| =

pqfor relatively prime positive integers p and q, find p + q

21 Consider a sequence x1, x2, · · · x12of real numbers such that x1= 1 and for n = 1, 2, , 10 let

xn+2= (xn+1+ 1)(xn+1− 1)

Suppose xn > 0 for n = 1, 2, , 11 and x12 = 0 Then the value of x2 can be written as

√ a+ √ b

positive integers a, b, c with a > b and no square dividing a or b Find 100a + 10b + c

2

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