tai lieu boi duong toan My 2

We denote by d1 the number of subset of S such that the sum of elements of the subset has remainder 7 when divided by 32.. We denote by d2the number of subset of S such that the sum of e

2005 MOSP Homework

1

Congratulations on your excellent performance on the AMC, AIME,and USAMO tests, which has earned you an invitation to attend theMath Olympiad Summer Program! This program will be an intenseand challenging opportunity for you to learn a tremendous amount ofmathematics

To better prepare yourself for MOSP, you need to work on thefollowing homework problems, which come from last year’s NationalOlympiads, from countries all around the world Even if some mayseem difficult, you should dedicate a significant amount of effort tothink about them—don’t give up right away All of you are highlytalented, but you may have a disappointing start if you do not put

in enough energy here At the beginning of the program, writtensolutions will be submitted for review by MOSP graders, and you willpresent your solutions and ideas during the first few study sessions.You are encouraged to use the email list to discuss these and otherinteresting math problems Also, if you have any questions aboutthese homework problems, please feel free to contact us

Zuming Feng, zfeng@exeter.eduMelania Wood, ¡lanie@Princeton.EDU¿

Problems for the Red Group

1

2 MOSP 2005 Homework Red Group

1.3 Let a, b, and c be real numbers Prove that

p

2(a2+ b2) +p2(b2+ c2) +p2(c2+ a2)

≥p3[(a + b)2+ (b + c)2+ (c + a)2].

1.4 Let x1, x2, , x5 be nonnegative real numbers such that x1+

x2 + x3 + x4 + x5 = 5 Determine the maximum value of

x1x2+ x2x3+ x3x4+ x4x5

1.5 Let S be a finite set of positive integers such that none of them

has a prime factor greater than three Show that the sum of the

reciprocals of the elements in S is smaller than three.

1.6 Find all function f : Z → R such that f (1) = 5/2 and that

f (x)f (y) = f (x + y) + f (x − y)

for all integers x and y.

1.7 Let x 1,1 , x 2,1 , , x n,1 , n ≥ 2, be a sequence of integers and assume that not all x i,1 are equal For k ≥ 2, if sequence {x i,k } n

MOSP 2005 Homework Red Group 3

(b) How many ways can eight mutually non-attacking rooks be

placed on the 9 × 9 chessboard so that all eight rooks are on

squares of the same color

1.9 Set S = {1, 2, , 2004} We denote by d1 the number of subset

of S such that the sum of elements of the subset has remainder

7 when divided by 32 We denote by d2the number of subset of

S such that the sum of elements of the subset has remainder 14

when divided by 16 Compute d1/d2

1.10 In a television series about incidents in a conspicuous town there

are n citizens staging in it, where n is an integer greater than 3.

Each two citizens plan together a conspiracy against one of theother citizens Prove that there exists a citizen, against whom atleast√ n other citizens are involved in the conspiracy.

1.11 Each of the players in a tennis tournament played one matchagainst each of the others If every player won at least one

match, show that there are three players A, B, and C such that

A beats B, B beats C, and C beats A Such a triple of player is

called a cycle Determine the number of maximum cycles such a

tournament can have

1.12 Determine if it is possible to choose nine points in the plane such

that there are n = 10 lines in the plane each of which passes through exactly three of the chosen points What if n = 11? 1.13 Let n be a positive integer Show that

1.14 A segment of length 2 is divided into n, n ≥ 2, subintervals A

square is then constructed on each subinterval Assume that thesum of the areas of all such squares is greater than 1 Show thatunder this assumption one can always choose two subintervalswith total length greater than 1

Geometry

1.15 Isosceles triangle ABC, with AB = AC, is inscribed in circle ω Point D lies on arc d BC not containing A Let E be the foot of

4 MOSP 2005 Homework Red Group

perpendicular from A to line CD Prove that BC + DC = 2DE 1.16 Let ABC be a triangle, and let D be a point on side AB Circle

ω1 passes through A and D and is tangent to line AC at A Circle ω2 passes through B and D and is tangent to line BC at

B Circles ω1and ω2 meet at D and E Point F is the reflection

of C across the perpendicular bisector of AB Prove that points

D, E, and F are collinear.

1.17 Let M be the midpoint of side BC of triangle ABC (AB > AC), and let AL be the bisector of the angle A The line passing through M perpendicular to AL intersects the side AB at the point D Prove that AD + M C is equal to half the perimeter of triangle ABC.

1.18 Let ABC be an obtuse triangle with ∠A > 90 ◦ , and let r and R

denote its inradius and circumradius Prove that

at P The bisector of angle BCA meet segments AD and BE at

Q and R, respectively Prove that

1.21 Let ABC be a triangle Prove that

1.22 Find all triples (x, y, z) in integers such that x2+ y2+ z2= 22004

1.23 Suppose that n is s positive integer Determine all the possible

values of the first digit after the decimal point in the decimalexpression of the number√ n3+ 2n2+ n.

MOSP 2005 Homework Red Group 5

1.24 Suppose that p and q are distinct primes and S is a subset of

{1, 2, , p−1} Let N (S) denote the number of ordered q-tuples

(x1, x2, , x q ) with x i ∈ S, 1 ≤ i ≤ q, such that

1.26 Find all ordered triple (a, b, c) of positive integers such that the

value of the expression

µ

b − 1a

¶ µ

c −1b

¶ µ

a −1c

¶

is an integer

1.27 Let a1 = 0, a2 = 1, and a n+2 = a n+1 + a n for all positive

integers n Show that there exists an increasing infinite arithmetic

progression of integers, which has no number in common in the

sequence {a n } n≥0

1.28 Let a, b, and c be pairwise distinct positive integers, which are

side lengths of a triangle There is a line which cuts both thearea and the perimeter of the triangle into two equal parts Thisline cuts the longest side of the triangle into two parts with ratio

2 : 1 Determine a, b, and c for which the product abc is minimal.

Problems for the Blue Group

7

8 MOSP 2005 Homework Blue Group

Algebra

2.1 Let a0, a1, a n be integers, not all zero, and all at least −1 Given a0+2a1+22a2+· · ·+2 n a n = 0, prove that a0+a1+· · ·+a n >

0

2.2 The sequence of real numbers {a n }, n ∈ N satisfies the following

condition: a n+1 = a n (a n + 2) for any n ∈ N Find all possible values for a2004

2.3 Determine all polynomials P (x) with real coefficients such that

(x3+ 3x2+ 3x + 2)P (x − 1) = (x3− 3x2+ 3x − 2)P (x) 2.4 Find all functions f : R → R such that

f (x3) − f (y3) = (x2+ xy + y2)(f (x) − f (y)).

2.5 Let a1, a2, , a2004be non-negative real numbers such that a1+

· · · + a2004≤ 25 Prove that among them there exist at least two

numbers a i and a j (i 6= j) such that

MOSP 2005 Homework Blue Group 9

Combinatorics

2.8 Consider all binary sequences (sequences consisting of 0’s and1’s) In such a sequence the following four types of operation are

allowed: (a) 010 → 1, (b) 1 → 010, (c) 110 → 0, and (d) 0 → 110.

Determine if it is possible to obtain sequence

2.9 Exactly one integer is written in each square of an n by n grid,

n ≥ 3 The sum of all of the numbers in any 2 × 2 square is

even and the sum of all the numbers in any 3 × 3 square is even Find all n for which the sum of all the numbers in the grid is

necessarily even

2.10 Let T be the set of all positive integer divisors of 2004100 What

is the largest possible number of elements that a subset S of T can have if no element of S is an integer multiple of any other element of S.

2.11 Consider an infinite array of integer Assume that each integer isequal to the sum of the integers immediately above and immedi-

ately to the left Assume that there exists a row R0such that all

the number in the row are positive Denote by R1 the row below

row R0, by R2 the row below row R1, and so on Show that for

each positive integer n, row R n cannot contain more than n zeros 2.12 Show that for nonnegative integers m and n,

2.13 A 10 × 10 × 10 cube is made up up from 500 white unit cubes

and 500 black unit cubes, arranged in such a way that every two

unit cubes that shares a face are in different colors A line is a

1 × 1 × 10 portion of the cube that is parallel to one of cube’s

edges From the initial cube have been removed 100 unit cubessuch that 300 lines of the cube has exactly one missing cube

10 MOSP 2005 Homework Blue Group

Determine if it is possible that the number of removed black unitcubes is divisible by 4

2.14 Let S be a set of points in the plane satisfying the following

conditions:

(a) there are seven points in S that form a convex heptagon; and (b) for any five points in S, if they form a convex pentagon, then there is point in S lies in the interior of the pentagon Determine the minimum value of the number of elements in S.

Geometry

2.15 Let ABCDEF be a equilateral convex hexagon with ∠A + ∠C +

∠E = ∠B + ∠D + ∠F Prove that lines AD, BE, and CF are

concurrent

2.16 Let ABCD be a convex quadrilateral Let P, Q be points on sides

BC and DC respectively such that ∠BAP = ∠DAQ Show that

the area of triangles ABP and ADQ is equal if and only if the

line through the orthocenters of these triangles is orthogonal to

2.18 The incircle O of an isosoceles triangle ABC with AB = AC meets BC, CA, AB at K, L, M respectively Let N be the inter- section of lines OL and KM and let Q be the intersection of lines

BN and CA Let P be the foot of the perpendicular from A to

BQ If we assume that BP = AP + 2P Q, what are the possible

values of AB/BC?

2.19 Let ABCD be a cyclic quadrilateral such that AB · BC = 2 · AD ·

DC Prove that its diagonals AC and BD satisfy the inequality

8BD2≤ 9AC2

2.20 A circle which is tangent to sides AB and BC of triangle ABC

is also tangent to its circumcircle at point T If I in the incenter

of triangle ABC, show that ∠AT I = ∠CT I.

MOSP 2005 Homework Blue Group 11

2.21 Points E, F, G, and H lie on sides AB, BC, CD, and DA of a convex quadrilateral ABCD such that

Points A, B, C, and D lie on sides H1E1, E1F1, F1G1, and G1H1

of a convex quadrilateral E1F1G1H1 such that E1F1 k EF ,

2.22 We call a natural number 3-partite if the set of its divisors can

be partitioned into 3 subsets each with the same sum Show thatthere exist infinitely many 3-partite numbers

2.23 Find all real numbers x such that

2.25 Find all prime numbers p and q such that 3p4+5p4+15 = 13p2q2.

2.26 Does there exist an infinite subset S of the natural numbers such that for every a, b ∈ S, the number (ab)2is divisible by a2−ab+b2?

2.27 A positive integer n is good if n can be written as the sum of 2004 positive integers a1, a2, , a2004 such that 1 ≤ a1 < a2< · · · <

a2004 and a i divides a i+1 for i = 1, 2, , 2003 Show that there

are only finitely many positive integers that are not good

2.28 Let n be a natural number and f1, f2, , f nbe polynomials with

integers coefficients Show that there exists a polynomial g(x)

which can be factored (with at least two terms of degree at least

1) over the integers such that f i (x) + g(x) cannot be factored

(with at least two terms of degree at least 1) over the integers for

every i.

Problems for the Black Group

13

14 MOSP 2005 Homework Black Group

3.4 Does there exist a function f : R → R such that for all x, y ∈ R,

holds for all natural numbers n.

3.6 Solve the system of equations

in the real numbers

3.7 Find all positive integers n for which there are distinct integers

3.8 Let X be a set with n elements and 0 ≤ k ≤ n Let a n,k

be the maximum number of permutations of the set X such that every two of them have at least k common components (where a common component of f and g is an x ∈ X such that

f (x) = g(x)) Let b n,k be the maximum number of permutations

MOSP 2005 Homework Black Group 15

of the set X such that every two of them have at most k common

components

(a) Show that a n,k · b n,k−1 ≤ n!.

(b) Let p be prime, and find the exact value of a p,2

3.9 A regular 2004-sided polygon is given, with all of its diagonalsdrawn After some sides and diagonals are removed, every vertexhas at most five segments coming out of it Prove that one cancolor the vertices with two colors such that at least 3/5 of theremaining segments have ends with different colors

3.10 Squares of an n × n table (n ≥ 3) are painted black and white as

in a chessboard A move allows one to choose any 2 × 2 square

and change all of its squares to the opposite color Find all such

n that there is a finite number of the moves described after which

all squares are the same color

3.11 A convex 2004-sided polygon P is given such that no four vertices are cyclic We call a triangle whose vertices are vertices of P thick

if all other 2001 vertices of P lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle Prove

that the number of thick triangles is equal to the number of thintriangles

3.12 A group consists of n tourists Among every three of them there

are at least two that are not familiar For any partition of thegroup into two busses, there are at least two familiar tourists inone bus Prove that there is a tourist who is familiar with at mosttwo fifth of all the tourists

3.13 A computer network is formed by connecting 2004 computers by

cables A set S of these computers is said to be independent if no pair of computers of S is connected by a cable Suppose that the

number of cables used is the minimum number possible such that

the size of any independent set is at most 50 Let c(L) be the number of cables connected to computer L Show that for any distinct computers A and B, c(A) = c(B) if they are connected

by a cable and |c(A)−c(B)| ≤ 1 otherwise Also, find the number

of cables used in the network

3.14 Eight problems were given to each of 30 students After thetest was given, point values of the problems were determined as

follows: a problem is worth n points if it is not solved by exactly

16 MOSP 2005 Homework Black Group

n contestants (no partial credit is given, only zero marks or full

marks)

(a) Is it possible that the contestant having got more points thatany other contestant had also solved less problems than anyother contestant?

(b) Is it possible that the contestant having got less points thanany other contestant has solved more problems than anyother contestant?

Geometry

3.15 A circle with center O is tangent to the sides of the angle with the vertex A at the points B and C Let M be a point on the larger of the two arcs BC of this circle (different from B and C) such that M does not lie on the line AO Lines BM and CM intersect the line AO at the points P and Q respectively Let K

be the foot of the perpendicular drawn from P to AC and L be the foot of the perpendicular drawn from Q to AB Prove that the lines OM and KL are perpendicular.

3.16 Let I be the incenter of triangle ABC, and let A1, B1, and C1be

arbitrary points lying on segments AI, BI, and CI, respectively The perpendicular bisectors of segments AA1, BB1, and CC1

form triangles A2B2C2 Prove that the circumcenter of triangle

A2B2C2coincides with the circumcenter of triangle ABC if and only if I is the orthocenter of triangle A1B1C1

3.17 Points M and M 0 are isogonal conjugates in the triangle ABC

(i.e ∠BAM = ∠M 0 AC, ∠ACM = ∠M 0 CB, ∠CBM =

∠M 0 BA) We draw perpendiculars M P , M Q, M R and M 0 P 0,

M 0 Q 0 , M 0 R 0 to the sides BC, AC, AB respectively Let QR, Q 0 R 0

and RP, R 0 P 0 and P Q, P 0 Q 0 intersect at E, F, G respectively Show that the lines EA, F B, and GC are parallel.

3.18 Let ABCD be a convex quadrilateral and let K, L, M, N be the midpoints of sides AB, BC, CD, DA respectively Let N L and

KM meet at point T Show that

8

3[DN T M ] < [ABCD] < 8[DN T M ],

where [P ] denotes area of P

MOSP 2005 Homework Black Group 17

3.19 Let ABCD be a cyclic quadrilateral such that AB · BC = 2 · AD ·

DC Prove that its diagonals AC and BD satisfy the inequality

8BD2≤ 9AC2

3.20 Given a convex quadrilateral ABCD The points P and Q are the midpoints of the diagonals AC and BD respectively The line

P Q intersects the lines AB and CD at N and M respectively.

Prove that the circumcircles of triangles N AP , N BQ, M QD, and M P C have a common point.

3.21 Let ABCD be a cyclic quadrilateral who interior angle at B is

60 degrees Show that if BC = CD, then CD + DA = AB Does

the converse hold?

Number Theory

3.22 Let n be a natural number and f1, f2, , f nbe polynomials with

integers coefficients Show that there exists a polynomial g(x)

which can be factored (with at least two terms of degree at least

1) over the integers such that f i (x) + g(x) cannot be factored

(with at least two terms of degree at least 1) over the integers for

every i.

3.23 Let a, b, c, and d be positive integers satisfy the following

proper-ties:

(a) there are exactly 2004 pairs of real numbers (x, y) with 0 ≤

x, y ≤ 1 such that both ax + by and cx + dy are integers

is written in the lowest form p n

q n ; that is p n and q n are relative

prime positive integers Find all n such that p n is divisible by 3

3.25 Prove that there does not exist an integer n > 1 such that n

divides 3n − 2 n

3.26 Find all integer solutions to

y2(x2+ y2− 2xy − x − y) = (x + y)2(x − y).

18 MOSP 2005 Homework Black Group

3.27 Let p be a prime number, and let 0 ≤ a1 < a2 < · · · < a m < p

and 0 ≤ b1< b2< · · · < b n < p be arbitrary integers Denote by

k the number of different remainders of a i + b j , 1 ≤ i ≤ m and

1 ≤ j ≤ n, modulo p Prove that

(i) if m + n > p, then k = p;

(ii) if m + n ≤ p, then k ≥ m + n − 1,

3.28 Let A be a finite subset of prime numbers and a be a positive integer Show that the number of positive integers m for which all prime divisors of a m − 1 are in A is finite.

Selected Problems from MOSP

2003 and 2004 Tests

20

MOSP 2005 Homework Selected Problems from MOSP 2003 and 2004 21

2+ 2ab (c + 2a)2+ (c + 2b)2 ≤ 1

µ

b + 2c

b + 2a

¶3+

4.4 Prove that for any nonempty finite set S, there exists a function

f : S × S → S satisfying the following conditions:

(a) for all a, b ∈ S, f (a, b) = f (b, a);

(b) for all a, b ∈ S, f (a, f (a, b)) = b;

(c) for all a, b, c, d ∈ S, f (f (a, b), f (c, d)) = f (f (a, c), f (b, d)).

4.5 Find all pairs (x, y) of real numbers with 0 < x < π

4.6 Prove that in any acute triangle ABC,

cot3A+cot3B+cot3C+6 cot A cot B cot C ≥ cot A+cot B+cot C.

4.7 Let a, b, c be positive real numbers Prove that

22 MOSP 2005 Homework Selected Problems from MOSP 2003 and 2004

4.9 Let A, B, C be real numbers in the interval (0, π

4.10 For a pair of integers a and b, with 0 < a < b < 1000, set

S ⊆ {1, 2, , 2003} is called a skipping set for (a, b) if for any

pair of elements s1, s2 ∈ S, |s1 − s2| 6∈ {a, b} Let f (a, b) be

the maximum size of a skipping set for (a, b) Determine the maximum and minimum values of f

4.11 Let R denote the set of real numbers Find all functions f : R →

R such that

f (x)f (yf (x) − 1) = x2f (y) − f (x)

for all real numbers x and y.

4.12 Show that there is an infinite sequence of positive integers

MOSP 2005 Homework Selected Problems from MOSP 2003 and 2004 23

4.18 Let N denote the set of positive integers, and let S be a set There

exists a function f : N → S such that if x and y are a pair of

positive integers with their difference being a prime number, then

f (x) 6= f (y) Determine the minimum number of elements in S.

4.19 Let n be a integer with n ≥ 2 Determine the number of

non-congruent triangles with positive integer side lengths two of which

sum to n.

4.20 Jess has 3 pegs and disks of different sizes Jess is supposed to

transfer the disks from one peg to another, and the disks have

to be sorted so that for any peg the disk at the bottom is the

largest on that peg (Discs above the bottom one may be in

any order.) There are n disks sorted from largest on bottom to

smallest on top at the start Determine the minimum number of

moves (moving one disk at a time) needed to move the disks to

another peg sorted in the same order

4.21 Let set S = {1, 2, , n} and set T be the set of all subsets of S

(including S and the empty set) One tries to choose three (not

necessarily distinct) sets from the set T such that either two of

the chosen sets are subsets of the third set or one of the chosen

set is a subset of both of the other two sets In how many ways

can this be done?

4.22 Let N denote the set of positive integers, and let f : N → N be

a function such that f (m) > f (n) for all m > n For positive