Đề olympic toán quốc tế international tại trung quốc national

Austria Federal Competition Part 1Austria Federal Competition For Advanced Students, Part 1 2002 1.. Austria Federal Competition For Advanced Students, Part 1 20041.. Austria Federal Com

Trang 1

A collection of National Mathematics Olympiads

July 19, 2014

Trang 2

Contents

Trang 3

Due to problems with the pdf generator at AoPS I decided to make my own collection of math Olympiadproblems in pdf format with links to problems at AoPS The idea is also that the list may be printed,and therefore a secondary aim is to not waste as much space as the AoPS pdf’s, but rather to start anew page only for the next country’s Olympiads Of course, when the document it printed, one will not

be able to follow links Therefore I include the link location in as minimalist a form I could think of Thelinks are all of the form http://www.artofproblemsolving.com/Forum/viewtopic.php?p=***,where *** is a number (the post number on AoPS), mostly consisting of 7 digits, but earlier posts mayhave less Thus after each problem that appears on AoPS (to my knowledge) I include the post number,which also links to the problem there

Of course, this is a work in progress, since there is quite a lot to do and there are constantly newcontests being written Therefore I start with the most popular contest, and the ones I have mostcomplete collections of Also, in stead of defining common terms over and over in problems that refer tothem, I include a glossary at the end, where undefined terms can be looked up

I also changed the margins in which the text is written This is not something I normally do, since theappearance turns out to be quite strange However, for printing purposes this saves a lot of pages This

is only a private collection and not something professional That is why I feel this change is warranted.There are also places where I have slightly altered the text This is mostly removing superfluousdefinitions like “where R is the set of real numbers” or “where bxc denotes the greatest integer ”, etc

Trang 4

Austria Federal Competition Part 1

Austria Federal Competition For Advanced Students, Part 1 2002

1 Determine all integers a and b such that

, where k goes over all integers k between 0 and 2002 whichare coprime to 2002

AoPS:2297681

4 Let A, C, P be three distinct points in the plane Construct all parallelograms ABCD such thatpoint P lies on the bisector of angle DAB and ∠AP D = 90◦

AoPS:2297685

1 Find all triples of prime numbers (p, q, r) such that pq+ pris a perfect square

4 In a parallelogram ABCD, points E and F are the midpoints of AB and BC, respectively, and

P is the intersection of EC and F D Prove that the segments AP, BP, CP and DP divide theparallelogram into four triangles whose areas are in the ratio 1 : 2 : 3 : 4

AoPS:2317094

Trang 5

1 Find all quadruples (a, b, c, d) of real numbers such that

a + bcd = b + cda = c + dab = d + abc

AoPS:2317197

2 A convex hexagon ABCDEF with AB = BC = a, CD = DE = b, EF = F A = c is inscribed

in a circle Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendiculardiagonals

AoPS:2317198

3 For natural numbers a, b, define Z(a, b) = (3a)!a!4·(4b)!·b! 3

(a) Prove that Z(a, b) is an integer for a ≤ b

(b) Prove that for each natural number b there are infinitely many natural numbers a such thatZ(a, b) is not an integer

1 Prove that there are infinitely many multiples of 2005 that contain all the digits 0, 1, 2, ,9 anequal number of times

AoPS:269328

2 For how many integers a with |a| ≤ 2005, does the system

x2 = y + a

y2 = x + ahave integer solutions?

AoPS:269331

3 For 3 real numbers a, b, c let sn= an+ bn+ cn

It is known that s1= 2, s2= 6 and s3= 14

Prove that for all natural numbers n > 1, we have |s2

n− sn −1sn+1| = 8

AoPS:269336

4 We are given two congruent, equilateral triangles ABC and P QR with parallel sides, but one hasone vertex pointing up and the other one has the vertex pointing down One is placed above theother so that the area of intersection is a hexagon A1A2A3A4A5A6 (labelled counterclockwise).Prove that A1A4, A2A5 and A3A6 are concurrent

AoPS:269346

Trang 6

1 Let n be a non-negative integer, which ends written in decimal notation on exactly k zeros, butwhich is bigger than 10k

For a n is only k = k(n) ≥ 2 known In how many different ways (as a function of k = k(n) ≥ 2)can n be written as difference of two squares of non-negative integers at least?

AoPS:1402125

2 Show that the sequence an= (n+1)7n2n+1n2−n is strictly increasing, where n = 0, 1, 2,

AoPS:1402086

3 In the triangle ABC let D and E be the boundary points of the incircle with the sides BC and

AC Show that if AD = BE holds, then the triangle is isosceles

AoPS:1402097

4 Given is the function f = bx2c + {x} for all x ∈ R+

Show that there exists an arithmetic sequence of different positive rational numbers, which all havethe denominator 3, if they are a reduced fraction, and don’t lie in the range of the function f

AoPS:1402126

1 In a quadratic table with 2007 rows and 2007 columns is an odd number written in each field.For 1 ≤ i ≤ 2007 is Zi the sum of the numbers in the i-th row and for 1 ≤ j ≤ 2007 is Sj the sum

of the numbers in the j-th column

A is the product of all Zi and B the product of all Sj

Show that A + B 6= 0

AoPS:1399124

2 For every positive integer n determine the highest value C(n), such that for every n-tuple (a1, a2, , an)

of pairwise distinct integers

4 Let n > 4 be a non-negative integer Given is the in a circle inscribed convex n-gon A0A1A2 An −1An

(An = A0) where the side Ai −1Ai = i (for 1 ≤ i ≤ n) Moreover, let φi be the angle between theline AiAi+1 and the tangent to the circle in the point Ai (where the angle φi is less than or equal

90o, i.e φiis always the smaller angle of the two angles between the two lines) Determine the sum

Trang 7

1 What is the remainder of the number

1 ·

20080

+ 2 ·

20081

+ · · · + 2009 ·

20082008

Show that there exists a sequence (an) in Fp with the property that for every other sequence (bn)

in Fp, the inequality an≤ bn holds for all n

4 In a triangle ABC let E be the midpoint of the side AC and F the midpoint of the side BC Let G

be the foot of the perpendicular from C to AB Show that 4EF G is isosceles if and only if 4ABC

is isosceles

1 Show that for all positive integer n the following inequality holds:

3n2 > (n!)4

2 For a positive integers n, k we define k-multifactorial of n as Fk(n) = n · (n − k) · (n − 2k) · · · r,where r is the reminder when n is divided by k that satisfy 1 ≤ r ≤ k Determine all non-negativeintegers n such that F20(n) + 2009 is a perfect square

3 There are n bus stops placed around the circular lake Each bus stop is connected by a road to thetwo adjacent stops (we call a segment the entire road between two stops) Determine the number

of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least onceand travel through exactly n + 1 segments

4 Let D, E, and F be respectively the midpoints of the sides BC, CA, and AB of 4ABC Let Ha,

Hb, Hc be the feet of perpendiculars from A, B, C to the opposite sides, respectively Let P , Q,

R be the midpoints of the HbHc, HcHa, and HaHb respectively Prove that P D, QE, and RF areconcurrent

1 Let f (n) =P2010

k=0 nk Show that for any integer m satisfying 2 ≤ m ≤ 2010, there exists no naturalnumber n such that f (n) is divisible by m

AoPS:1874739

Trang 8

2 For a positive integer n, we define the function fn(x) =Pn

k=1|x − k| for all real numbers x For anytwo-digit number n (in decimal representation), determine the set of solutions Ln of the inequality

fn(x) < 41

AoPS:1874750

3 Given is the set Mn = {0, 1, 2, , n} of non-negative integers less than or equal to n A subset S

of Mn is called outstanding if it is non-empty and for every natural number k ∈ S, there exists ak-element subset Tk of S

Determine the number a(n) of outstanding subsets of Mn

1 Determine all integer triplets (x, y, z) such that x4+ x2= 7zy2

of the other two

How many three-element subsets of the set of integers {z ∈ Z | −2011 < z < 2011} are arithmeticand harmonic?

AoPS:2270919

4 Inside or on the faces of a tetrahedron with five edges of length 2 and one edge of length 1, there

is a point P having distances a, b, c, d to the four faces of the tetrahedron Determine the locus ofall points P such that a + b + c + d is minimal and the locus of all points P such that a + b + c + d

is maximal

AoPS:2270922

1 Determine all functions f : Z → Z satisfying the following property: For each pair of integers mand n (not necessarily distinct), gcd(m, n) divides f (m) + f (n)

AoPS:2693139

2 Determine all solutions (n, k) of the equation n! + An = nk

with n, k ∈ N for A = 7 and for

A = 2012

AoPS:2693137

Trang 9

3 Consider a stripe of n fields, numbered from left to right with the integers 1 to n in ascendingorder Each of the fields is coloured with one of the colours 1, 2 or 3 Even-numbered fields can

be coloured with any colour Odd-numbered fields are only allowed to be coloured with the oddcolours 1 and 3 How many such colourings are there such that any two neighbouring fields havedifferent colours?

AoPS:2693138

4 Let ABC be a scalene (i.e non-isosceles) triangle Let U be the centre of the circumcircle of thistriangle and I the centre of the incircle Assume that the second point of intersection differentfrom C of the angle bisector of γ = ∠ACB with the circumcircle of ABC lies on the perpendicularbisector of U I Show that γ is the second-largest angle in the triangle ABC

AoPS:2693136

1 Show that if for non-negative integers m, n, N , k the equation

AoPS:3103104

3 Arrange the positive integers into two lines as follows: 1 We start with writing 1 in the upper line,

2 in the lower line and 3 again in the upper line Afterwards, we alternately write one single integer

in the upper line and a block of integers in the lower line The number of consecutive integers in ablock is determined by the first number in the previous block

Let a1, a2, a3, be the numbers in the upper line Give an explicit formula for an

AoPS:3103138

4 Let A, B and C be three points on a line (in this order) For each circle k through the points Band C, let D be one point of intersection of the perpendicular bisector of BC with the circle k.Further, let E be the second point of intersection of the line AD with k Show that for each circle

k, the ratio of lengths BE : CE is the same

AoPS:3103150

Trang 10

Austria Federal Competition Part 2

1 Let a be a fixed integer Find all integer solutions x, y, z of the system

5x + (a + 2)y + (a + 2)z = a,(2a + 4)x + (a2+ 3)y + (2a + 2)z = 3a − 1,(2a + 4)x + (2a + 2)y + (a2+ 3)z = a + 1

AoPS:2344288

2 A positive integer K is given Define the sequence (an) by a1= 1 and anis the n-th positive integergreater than an −1 which is congruent to n modulo K

(a) Find an explicit formula for an

(b) What is the result if K = 2?

AoPS:2344289

3 Let be given a triangle ABC Points P on side AC and Y on the production of CB beyond B arechosen so that Y subtends equal angles with AP and P C Similarly, Q on side BC and X on theproduction of AC beyond C are such that X subtends equal angles with BQ and QC Lines Y Pand XB meet at R, XQ and Y A meet at S, and XB and Y A meet at D Prove that P QRS is aparallelogram if and only if ACBD is a cyclic quadrilateral

AoPS:2344294

4 Determine all quadruples (a, b, c, d) of real numbers satisfying the equation

256a3b3c3d3= (a6+ b2+ c2+ d2)(a2+ b6+ c2+ d2)(a2+ b2+ c6+ d2)(a2+ b2+ c2+ d6)

AoPS:2344299

5 We define the following operation which will be applied to a row of bars being situated side-by-side

on positions 1, 2, , N Each bar situated at an odd numbered position is left as is, while each bar

at an even numbered position is replaced by two bars After that, all bars will be put side-by- side

in such a way that all bars form a new row and are situated on positions 1, , M From an initialnumber a0 > 0 of bars there originates a sequence (an)n ≥0, where an is the number of bars afterhaving applied the operation n times

(a) Prove that for no n > 0 can we have an= 1997

(b) Determine all natural numbers that can only occur as a0or a1

AoPS:2344302

6 For every natural number n, find all polynomials x2+ax+b, where a2≥ 4b, that divide x2n+axn+b

AoPS:2344303

Trang 11

1 Let a ≥ 0 be a natural number Determine all rational x, so that

Qn, (b) Kn or (c) QnKn divides? If one divides Q98K98 by the highest power of 98, then one get

a number N By which power-of-two number is N still divisible?

4 Let M be the set of the vertices of a regular hexagon, our Olympiad symbol How many chains

∅ ⊂ A ⊂ B ⊂ C ⊂ D ⊂ M of six different set, beginning with the empty set and ending with the

M , are there?

AoPS:2336861

5 Let P (x) = x3− px2+ qx − r be a cubic polynomial with integer roots a, b, c

(a) Show that the greatest common divisor of p, q, r is equal to 1 if the greatest common divisor

AoPS:2336877

1 Prove that for each positive integer n, the sum of the numbers of digits of 4n and of 25n (in thedecimal system) is odd

AoPS:2336031

4 Ninety-nine points are given on one of the diagonals of a unit square Prove that there is at mostone vertex of the square such that the average squared distance from a given point to the vertex isless than or equal to 1/2

AoPS:2336037

Trang 12

5 Given a real number A and an integer n with 2 ≤ n ≤ 19, find all polynomials P (x) with realcoefficients such that P (P (P (x))) = Axn+ 19x + 99.

AoPS:2336041

6 Two players A and B play the following game An even number of cells are placed on a circle Abegins and A and B play alternately, where each move consists of choosing a free cell and writingeither O or M in it The player after whose move the word OM O occurs for the first time in threesuccessive cells wins the game If no such word occurs, then the game is a draw Prove that ifplayer B plays correctly, then player A cannot win

AoPS:2336045

1 The sequence an is defined by a0= 4, a1= 1 and the recurrence formula an+1= an+ 6an −1 Thesequence bn is given by

AoPS:2335538

5 Find all pairs of integers (m, n) such that

(m2+ 2000m + 999999) − (3n3+ 9n2+ 27n) = 1

Trang 13

2 Determine all triples of positive real numbers (x, y, z) such that

x + y + z = 6,1

AoPS:2330086

3 A triangle ABC is inscribed in a circle with centre U and radius r A tangent c0 to a larger circleK(U, 2r) is drawn so that C lies between the lines c = AB and C0 Lines a0 and b0 are analogouslydefined The triangle formed by a0, b0, c0 is denoted A0B0C0 Prove that the three lines, joining themidpoints of pairs of parallel sides of the two triangles, have a common point

AoPS:2330099

1 Consider all possible rectangles that can be drawn on a 8 × 8 chessboard, covering only whole cells.Calculate the sum of their areas

What formula is obtained if “8 × 8” is replaced with “a × b”, where a, b are positive integers?

4 Find all polynomials P (x) of the smallest possible degree with the following properties:

(i) The leading coefficient is 200; (ii) The coefficient at the smallest non-vanishing power is 2; (iii)The sum of all the coefficients is 4; (iv) P (−1) = 0, P (2) = 6, P (3) = 8

AoPS:2330759

5 In the net drawn below, in how many ways can one reach the point 3n + 1 starting from the point

1 so that the labels of the points on the way increase?

AoPS:2330771

6 Let H be the orthocentre of an acute-angled triangle ABC Show that the triangles ABH, BCHand CAH have the same perimeter if and only if the triangle ABC is equilateral

AoPS:2330774

Trang 14

34

11

1 Consider the polynomial P (n) = n3− n2− 5n + 2 Determine all integers n for which P (n)2 is asquare of a prime

5 We are given sufficiently many stones of the forms of a rectangle 2 × 1 and square 1 × 1 Let n > 3

be a natural number In how many ways can one tile a rectangle 3 × n using these stones, so that

no two 2 × 1 rectangles have a common point, and each of them has the longer side parallel to theshorter side of the big rectangle?

AoPS:2317108

6 Let ABC be an acute-angled triangle The circle k with diameter AB intersects AC and BC again

at P and Q, respectively The tangents to k at A and Q meet at R, and the tangents at B and Pmeet at S Show that C lies on the line RS

AoPS:2317110

1 Prove without using advanced (differential) calculus:

(a) For any real numbers a, b, c, d it holds that a6+ b6+ c6+ d66abcd ≥ 2 When does equalityhold?

(b) For which natural numbers k does some inequality of the form ak+ bk+ ck+ dkkabcd ≥ Mk

hold for all real a, b, c, d? For each such k, find the greatest possible value of Mk and determinethe cases of equality

2 (a) Given any set {p1, p2, , pk} of prime numbers, show that the sum of the reciprocals of all

numbers of the form pr 1

1 · · · prk

k (r1, , rk ∈ N) is also a reciprocal of an integer

(b) Compute the above sum, knowing that 1/2004 occurs among the summands

(c) Prove that for each k-element set {p1, , pk} of primes (k > 2), the above sum is smallerthan 1/N , where N = 2 · 3k2(k2)!

Trang 15

3 A trapezoid ABCD with perpendicular diagonals AC and BD is inscribed in a circle k Let kaand

kc respectively be the circles with diameters AB and CD Compute the area of the region which isinside the circle k, but outside the circles ka and kc

4 Show that there is an infinite sequence a1, a2, of natural numbers such that a2+ a2+ · · · + a2

N

is a perfect square for all N Give a recurrent formula for one such sequence

5 Solve the following system of equations in real numbers:

a2=

√

bc√3

bcd(b + c)(b + c + d), b

2=

√

cd√3

cda(c + d)(c + d + a),

c2=

√

da√3

dab(d + a)(d + a + b), d

2=

√

ab√3

abc(a + b)(a + b + c).

6 Outside an equilateral triangle ABC of area 1, three triangles BCP , CAQ, ABR are constructed

so that ∠P = ∠Q = ∠R = 60◦

(a) What is the greatest possible area of triangle P QR?

(b) What is the greatest possible area of the triangle whose vertices are the incentres of trianglesBCP , CAQ, ABR?

1 Find all triples (a, b, c) of natural numbers, such that lcm(a, b, c) = a + b + c

AoPS:269348

2 Prove that for all a, b, c, d ∈ R+, we have

a + b + c + dabcd ≤

3 Triangle DEF is acute Circle c1 is drawn with DF as its diameter and circle c2is drawn with DE

as its diameter Points Y and Z are on DF and DE respectively so that EY and F Z are altitudes

of triangle DEF EY intersects c1 at P , and F Z intersects c2at Q EY extended intersects c1at

R, and F Z extended intersects c2 at S Prove that P , Q, R, and S are concyclic points

AoPS:269364

4 The function f : {0, , 2005} → N has the properties that f (2x + 1) = f (2x), f (3x + 1) = f (3x)and f (5x + 1) = f (5x) with x ∈ {0, 1, 2, , 2005} How many different values can the functionassume?

AoPS:269379

5 Find all real a, b, c, d, e, f that satisfy the system

4a = (b + c + d + e)4

4b = (c + d + e + f )44c = (d + e + f + a)44d = (e + f + a + b)44e = (f + a + b + c)44f = (a + b + c + d)4

AoPS:269382

6 Let Q be a point inside a cube Prove that there are infinitely many lines l so that AQ = BQ where

A and B are the two points of intersection of l and the surface of the cube

AoPS:269387

Trang 16

1 Let N be a positive integer How many non-negative integers n ≤ N are there that have an integermultiple, that only uses the digits 2 and 6 in decimal representation?

3 The triangle ABC is given On the extension of the side AB we construct the point R with

BR = BC, where AR > BR and on the extension of the side AC we construct the point S with

CS = CB, where AS > CS Let A1be the point of intersection of the diagonals of the quadrilateralBRSC

Analogous we construct the point T on the extension of the side BC, where CT = CA and BT > CTand on the extension of the side BA we construct the point U with AU = AC, where BU > AU Let B1 be the point of intersection of the diagonals of the quadrilateral CT U A

Likewise we construct the point V on the extension of the side CA, where AV = AB and CV > AVand on the extension of the side CB we construct the point W with BW = BA and CW > BW Let C1 be the point of intersection of the diagonals of the quadrilateral AV W B

Show that the area of the hexagon AC1BA1CB1 is equal to the sum of the areas of the trianglesABC and A1B1C1

1 For which non-negative integers a < 2007 the congruence x2+ a ≡ 0 mod 2007 has got exactly twodifferent non-negative integer solutions?

That means, that there exist exactly two different non-negative integers u and v less than 2007,such that u2+ a and v2+ a are both divisible by 2007

AoPS:1399005

Trang 17

2 Find all tuples (x1, x2, x3, x4, x5, x6) of non-negative integers, such that the following system ofequations holds:

AoPS:1399018

5 Given is a convex n-gon with a triangulation, that is a partition into triangles through diagonalsthat don’t cut each other Show that it is always possible to mark the n corners with the digits ofthe number 2007 such that every quadrilateral consisting of 2 neighboured (along an edge) triangleshas got 9 as the sum of the numbers on its 4 corners

AoPS:1399037

6 The triangle ABC with the circumcircle k(U, r) is given On the extension of the radii U A a point

P is chosen The reflection of the line P B on the line BA is called g Likewise the reflection of theline P C on the line CA is called h The intersection of g and h is called Q

Find the geometric location of all possible intersections Q, while P passes through the extension ofthe radii U A

AoPS:1399021

1 Prove the inequality

√

a1−ab1−bc1−c≤1

3holds for all positive real numbers a, b and c with a + b + c = 1

AoPS:1397365

2 (a) Does there exist a polynomial P (x) with coefficients in integers, such that P (d) = 2008d holds

for all positive divisors of 2008?

(b) For which positive integers n does a polynomial P (x) with coefficients in integers exists, suchthat P (d) = n

d holds for all positive divisors of n?

Trang 18

6 We are given a square ABCD Let P be a point not equal to a corner of the square or to its centre

M For any such P , we let E denote the common point of the lines P D and AC, if such a pointexists Furthermore, we let F denote the common point of the lines P C and BD, if such a pointexists All such points P , for which E and F exist are called acceptable points Determine the set

of all acceptable points, for which the line EF is parallel to AD

Determine all pairs (x, y) of non-negative integers, dependent on k > 0, such that Akk(x) = Bk(y)

2 (a) For positive integers a < b let

M (a, b) =

Pb k=a

√

k2+ 3k + 3

b − a + 1 .Calculate bM (a, b)c

(b) Calculate

N (a, b) =

Pb k=a

4 Let a be a positive integer Consider the sequence (an) defined as a0= a and an = an1+ 40n! for

n > 0 Prove that the sequence (an) has infinitely many numbers divisible by 2009

5 Let n > 1 and for 1 ≤ k ≤ n let pk = pk(a1, a2, , an) be the sum of the products of all possiblecombinations of k of the numbers a1, a2, , an Furthermore let P = P (a1, a2, , an) be the sum

of all pk with odd values of k less than or equal to n

How many different values are taken by aj if all the numbers aj (1 ≤ j ≤ n) and P are prime?

6 The quadrilateral P QRS whose vertices are the midpoints of the sides AB, BC, CD, DA, tively of a quadrilateral ABCD is called the midpoint quadrilateral of ABCD

respec-Determine all circumscribed quadrilaterals whose mid-point quadrilaterals are squares

Trang 19

1 Show that

(x − y)7+ (y − z)7+ (z − x)7− (x − y)(y − z)(z − x) (x − y)4+ (y − z)4+ (z − x)4

(x − y)5+ (y − z)5+ (z − x)5 ≥ 3holds for all triples of distinct integers x, y, z When does equality hold?

2 Determine all triples (x, y, z) of positive integers x > y > z > 0, such that x2= y · 2z+ 1

3 On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle

at the point of impact A regular hexagon with its vertices on the circle is drawn on a circularbilliard table A (point-shaped) ball is placed somewhere on the circumference of the hexagon, butnot on one of its edges Describe a periodical track of this ball with exactly four points at the rails.With how many different directions of impact can the ball be brought onto such a track?

4 Consider the part of a lattice given by the corners (0, 0), (n, 0), (n, 2) and (0, 2) From a latticepoint (a, b) one can move to (a + 1, b) or to (a + 1, b + 1) or to (a, b − 1), provided that the secondpoint is also contained in the part of the lattice

How many ways are there to move from (0, 0) to (n, 2) considering these rules?

5 Two decompositions of a square into three rectangles are called substantially different, if reorderingthe rectangles does not change one into the other

How many substantially different decompositions of a 2010 × 2010 square in three rectangles withinteger side lengths are there such that the area of one rectangle is equal to the arithmetic mean ofthe areas of the other rectangles?

6 A diagonal of a convex hexagon is called “long” if it decomposes the hexagon into two quadrangles.Each pair of long diagonals decomposes the hexagon into two triangles and two quadrangles Given

is a hexagon with the property, that for each decomposition by two long diagonals the resultingtriangles are both isosceles with the side of the hexagon as base

Show that the hexagon has a circumcircle

1 Every brick has 5 holes in a line The holes can be filled with bolts (fitting in one hole) and braces(fitting into two neighbouring holes) No hole may remain free One puts n of these bricks in a line

to form a pattern from left to right In this line no two braces and no three bolts may be adjacent.How many different such patterns can be produced with n bricks?

AoPS:2297333

2 We consider permutations f of the set N of non-negative integers, i.e bijective maps f from N to

N, with the following additional properties: n > 42 Further, for all integers n > 42, K Show thatthere are infinitely many natural numbers K such that f maps the set {n | 0 ≤ n ≤ K} onto itself

AoPS:2297335

3 We are given a non-isosceles triangle ABC with incentre I Show that the circumcircle k of thetriangle AIB does not touch the lines CA and CB Let P be the second point of intersection of kwith CA and let Q be the second point of intersection of k with CB Show that the four points A,

B, P and Q (not necessarily in this order) are the vertices of a trapezoid

AoPS:2297336

4 Determine all pairs (a, b) of non-negative integers, such that ab + b divides a2b + 2b (Remark:

00= 1.)

AoPS:2297337

Trang 20

5 Let k and n be positive integers Show that if xj(1 ≤ j ≤ n) are real numbers withPn

j=1 1

6 Two circles k1 and k2 with radii r1 and r2 touch each outside at point Q The other endpoints

of the diameters through Q are P on k1 and R on k2 We choose two points A and B, one oneach of the arcs P Q of k1 (P BQA is a convex quadrangle.) Further, let C be the second point

of intersection of the line AQ with k2 and let D be the second point of intersection of the line BQwith k2 The lines P B and RC intersect in U and the lines P A and RD intersect in V Show thatthere is a point Z that lies on all of these lines U V

AoPS:2297343

1 Determine the maximum value of m, such that the inequality

(a2+ 4(b2+ c2))(b2+ 4(a2+ c2))(c2+ 4(a2+ b2)) ≥ mholds for every a, b, c ∈ R \ {0} with 1

a

+ 1 b

Định dạng
Số trang	743
Dung lượng	5,39 MB