Tài liệu Tuyển Tập Các Đề Thi Của Các Nước Trên Thế Giới P2 pptx

We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends.. Find all positive integers with two or more digits such that if we insert a

A1 1999 cards are lying on a table Each card has a red side and a black side and can be

either side up Two players play alternately Each player can remove any number of cards

showing the same color from the table or turn over any number of cards of the same color

The winner is the player who removes the last card Does the first or second player have a

winning strategy?

A2 Show that there is no arithmetic progression of 1999 distinct positive primes all less than

12345

A3 P is a point inside the triangle ABC D, E, F are the midpoints of AP, BP, CP The lines

BF, CE meet at L; the lines CD, AF meet at M; and the lines AE, BD meet at N Show that

area DNELFM = (1/3) area ABC Show that DL, EM, FN are concurrent

B1 10 squares of a chessboard are chosen arbitrarily and the center of each chosen square is

marked The side of a square of the board is 1 Show that either two of the marked points are a

distance ≤ √2 apart or that one of the marked points is a distance 1/2 from the edge of the

board

B2 ABCD has AB parallel to CD The exterior bisectors of ∠B and ∠C meet at P, and the

B3 A polygon has each side integral and each pair of adjacent sides perpendicular (it is not

necessarily convex) Show that if it can be covered by non-overlapping 2 x 1 dominos, then at

least one of its sides has even length

14th Mexican 2000

A1 A, B, C, D are circles such that A and B touch externally at P, B and C touch externally

at Q, C and D touch externally at R, and D and A touch externally at S A does not intersect

C, and B does not intersect D Show that PQRS is cyclic If A and C have radius 2, B and D

have radius 3, and the distance between the centers of A and C is 6, find area PQRS

A2 A triangle is constructed like that below, but with 1, 2, 3, , 2000 as the first row Each

number is the sum of the two numbers immediately above Find the number at the bottom of

A3 If A is a set of positive integers, take the set A' to be all elements which can be written as

± a1 ± a2 ± an, where ai are distinct elements of A Similarly, form A" from A' What is the

smallest set A such that A" contains all of 1, 2, 3, , 40?

B1 Given positive integers a, b (neither a multiple of 5) we construct a sequence as follows:

first composite member of the sequence?

B2 Given an n x n board with squares colored alternately black and white like a chessboard

An allowed move is to take a rectangle of squares (with one side greater than one square, and

both sides odd or both sides even) and change the color of each square in the rectangle For

which n is it possible to end up with all the squares the same color by a sequence of allowed

moves?

B3 ABC is a triangle with ∠B > 90o H is a point on the side AC such that AH = BH and

BH is perpendicular to BC D, E are the midpoints of AB, BC The line through H parallel to

AB meets DE at F Show that ∠BCF = ∠ACD

A1 Find all 7-digit numbers which are multiples of 21 and which have each digit 3 or 7

A2 Given some colored balls (at least three different colors) and at least three boxes The

balls are put into the boxes so that no box is empty and we cannot find three balls of different

colors which are in three different boxes Show that there is a box such that all the balls in all

the other boxes have the same color

A3 ABCD is a cyclic quadrilateral M is the midpoint of CD The diagonals meet at P The

circle through P which touches CD at M meets AC again at R and BD again at Q The point S

on BD is such that BS = DQ The line through S parallel to AB meets AC at T Show that AT

= RC

B1 For positive integers n, m define f(n,m) as follows Write a list of 2001 numbers ai,

where a1 = m, and ak+1 is the residue of ak2 mod n (for k = 1, 2, , 2000) Then put f(n,m) = a1

- a2 + a3 - a4 + a5 - + a2001 For which n ≥ 5 can we find m such that 2 ≤ m ≤ n/2 and f(m,n)

> 0?

B2 ABC is a triangle with AB < AC and ∠A = 2 ∠C D is the point on AC such that CD =

and the line through C parallel to AB at N Show that MD = ND

B3 A collector of rare coins has coins of denominations 1, 2, , n (several coins for each

denomination) He wishes to put the coins into 5 boxes so that: (1) in each box there is at

most one coin of each denomination; (2) each box has the same number of coins and the same

denomination total; (3) any two boxes contain all the denominations; (4) no denomination is

in all 5 boxes For which n is this possible?

16th Mexican 2002

A1 The numbers 1 to 1024 are written one per square on a 32 x 32 board, so that the first

row is 1, 2, , 32, the second row is 33, 34, , 64 and so on Then the board is divided into

four 16 x 16 boards and the position of these boards is moved round clockwise, so that AB

goes to DA, DC goes to CB

then each of the 16 x 16 boards is divided into four equal 8 x 8 parts and each of these is

moved around in the same way (within the 16 x 16 board) Then each of the 8 x 8 boards is

divided into four 4 x 4 parts and these are moved around, then each 4 x 4 board is divided into

2 x 2 parts which are moved around, and finally the squares of each 2 x 2 part are moved

around What numbers end up on the main diagonal (from the top left to bottom right)?

A2 ABCD is a parallelogram K is the circumcircle of ABD The lines BC and CD meet K

again at E and F Show that the circumcenter of CEF lies on K

A3 Does n2 have more divisors = 1 mod 4 or = 3 mod 4?

B1 A domino has two numbers (which may be equal) between 0 and 6, one at each end The

domino may be turned around There is one domino of each type, so 28 in all We want to

form a chain in the usual way, so that adjacent dominos have the same number at the adjacent

ends Dominos can be added to the chain at either end We want to form the chain so that after

each domino has been added the total of all the numbers is odd For example, we could place

first the domino (3,4), total 3 + 4 = 7 Then (1,3), total 1 + 3 + 3 + 4 = 11, then (4,4), total 11

+ 4 + 4 = 19 What is the largest number of dominos that can be placed in this way? How

many maximum-length chains are there?

B2 A trio is a set of three distinct integers such that two of the numbers are divisors or

multiples of the third Which trio contained in {1, 2, , 2002} has the largest possible sum?

Find all trios with the maximum sum

B3 ABCD is a quadrilateral with ∠A = ∠B = 90o M is the midpoint of AB and ∠CMD =

A1 Find all positive integers with two or more digits such that if we insert a 0 between the

units and tens digits we get a multiple of the original number

A2 A, B, C are collinear with B betweeen A and C K1 is the circle with diameter AB, and

and CS meet on the perpendicular to AC at B

A3 At a party there are n women and n men Each woman likes r of the men, and each man

likes r of then women For which r and s must there be a man and a woman who like each

other?

B1 The quadrilateral ABCD has AB parallel to CD P is on the side AB and Q on the side

CD such that AP/PB = DQ/CQ M is the intersection of AQ and DP, and N is the intersection

of PC and QB Find MN in terms of AB and CD

B2 Some cards each have a pair of numbers written on them There is just one card for each

pair (a,b) with 1 ≤ a < b ≤ 2003 Two players play the following game Each removes a card

in turn and writes the product ab of its numbers on the blackboard The first player who

causes the greatest common divisor of the numbers on the blackboard to fall to 1 loses Which

player has a winning strategy?

B3 Given a positive integer n, an allowed move is to form 2n+1 or 3n+2 The set Sn is the set

of all numbers that can be obtained by a sequence of allowed moves starting with n For

compatible if Sm ∩ Sn is non-empty Which members of {1, 2, 3, , 2002} are compatible

with 2003?

Polish (1983 – 2003)

A1 The angle bisectors of the angles A, B, C in the triangle ABC meet the circumcircle again

at K, L, M Show that |AK| + |BL| + |CM| > |AB| + |BC| + |CA|

A2 For given n, we choose k and m at random subject to 0 ≤ k ≤ m ≤ 2n Let pn be the

probability that the binomial coefficient mCk is even Find limn→∞ pn

A3 Q is a point inside the n-gon P1P2 Pn which does not lie on any of the diagonals Show

that if n is even, then Q must lie inside an even number of triangles PiPjPk

B1 Given a real numbers x ∈ (0,1) and a positive integer N, prove that there exist positive

integers a, b, c, d such that (1) a/b < x < c/d, (2) c/d - a/b < 1/n, and (3) qr - ps = 1

B2 There is a piece in each square of an m x n rectangle on an infinite chessboard An

allowed move is to remove two pieces which are adjacent horizontally or vertically and to

place a piece in an empty square adjacent to the two removed and in line with them (as shown

35th Polish 1984

A1 X is a set with n > 2 elements Is there a function f : X → X such that the composition f

n-1 is constant, but f n-2 is not constant?

A2 Given n we define ai,j as follows For i, j = 1, 2, , n, ai,j = 1 for j = i, and 0 for j ≠ i For

i = 1, 2, , n, j = n+1, , 2n, ai,j = -1/n Show that for any permutation p of (1, 2, , 2n) we

have ∑i=1n |∑k=1n ai,p(k) | ≥ n/2

A3 W is a regular octahedron with center O P is a plane through the center O K(O, r1) and

K(O, r2) are circles center O and radii r1, r2 such that K(O, r1) P∩W K(O, r2) Show that

r1/r2 ≤ (√3)/2

B1 We throw a coin n times and record the results as the sequence α1, α2, , αn, using 1 for

head, 2 for tail Let βj = α1 + α2 + + αj and let p(n) be the probability that the sequence β1,

β2, , βn includes the value n Find p(n) in terms of p(n-1) and p(n-2)

B2 Six disks with diameter 1 are placed so that they cover the edges of a regular hexagon

with side 1 Show that no vertex of the hexagon is covered by two or more disks

B3 There are 1025 cities, P1, , P1025 and ten airlines A1, , A10, which connect some of the

cities Given any two cities there is at least one airline which has a direct flight between them

Show that there is an airline which can offer a round trip with an odd number of flights

A1 Find the largest k such that for every positive integer n we can find at least k numbers in

the set {n+1, n+2, , n+16} which are coprime with n(n+17)

A2 Given a square side 1 and 2n positive reals a1, b1, , an, bn each ≤ 1 and satisfying ∑ aibi

parallel to the square sides

A3 The function f : R → R satisfies f(3x) = 3f(x) - 4f(x)3 for all real x and is continuous at x

= 0 Show that |f(x)| ≤ 1 for all x

B1 P is a point inside the triangle ABC is a triangle The distance of P from the lines BC,

CA, AB is da, db, dc respectively Show that 2/(1/da + 1/db + 1/dc) < r < (da + db + dc)/2, where

r is the inradius

B2 p(x,y) is a polynomial such that p(cos t, sin t) = 0 for all real t Show that there is a

polynomial q(x,y) such that p(x,y) = (x2 + y2 - 1) q(x,y)

B3 There is a convex polyhedron with k faces Show that if k/2 of the faces are such that no

two have a common edge, then the polyhedron cannot have an inscribed sphere

37th Polish 1986

A1 A square side 1 is covered with m2 rectangles Show that there is a rectangle with

perimeter at least 4/m

A2 Find the maximum possible volume of a tetrahedron which has three faces with area 1

A3 p is a prime and m is a non-negative integer < p-1 Show that ∑j=1p jm is divisible by p

B1 Find all n such that there is a real polynomial f(x) of degree n such that f(x) ≥ f '(x) for all

real x

B2 There is a chess tournament with 2n players (n > 1) There is at most one match between

each pair of players If it is not possible to find three players who all play each other, show

is possible to arrange them so that we cannot find three players who all play each other

B3 ABC is a triangle The feet of the perpendiculars from B and C to the angle bisector at A

are K, L respectively N is the midpoint of BC, and AM is an altitude Show that K,L,N,M are

concyclic

A1 There are n ≥ 2 points in a square side 1 Show that one can label the points P1, P2, , Pn

such that ∑i=1n |Pi-1 - Pi|2 ≤ 4, where we use cyclic subscripts, so that P0 means Pn

A2 A regular n-gon is inscribed in a circle radius 1 Let X be the set of all arcs PQ, where P,

two or more of the Li can be the same) Show that the expected length of L1 ∩ L2 ∩ L3 ∩ L4 ∩

L5 is independent of n

A3 w(x) is a polynomial with integral coefficients Let pn be the sum of the digits of the

number w(n) Show that some value must occur infinitely often in the sequence p1, p2, p3,

B1 Let S be the set of all tetrahedra which satisfy (1) the base has area 1, (2) the total face

area is 4, and (3) the angles between the base and the other three faces are all equal Find the

element of S which has the largest volume

B2 Find the smallest n such that n2-n+11 is the product of four primes (not necessarily

distinct)

B3 A plane is tiled with regular hexagons of side 1 A is a fixed hexagon vertex Find the

number of paths P such that (1) one endpoint of P is A, (2) the other endpoint of P is a

hexagon vertex, (3) P lies along hexagon edges, (4) P has length 60, and (5) there is no shorter

path along hexagon edges from A to the other endpoint of P

39th Polish 1988

A1 The real numbers x1, x2, , xn belong to the interval (0,1) and satisfy x1 + x2 + + xn =

m + r, where m is an integer and r ∈ [0,1) Show that x12 + x22 + + xn2 ≤ m + r2

A2 For a permutation P = (p1, p2, , pn) of (1, 2, , n) define X(P) as the number of j such

that pi < pj for every i < j What is the expected value of X(P) if each permutation is equally

likely?

A3 W is a polygon W has a center of symmetry S such that if P belongs to W, then so does

P', where S is the midpoint of PP' Show that there is a parallelogram V containing W such

that the midpoint of each side of V lies on the border of W

B1 d is a positive integer and f : [0,d] → R is a continuous function with f(0) = f(d) Show

that there exists x ∈ [0,d-1] such that f(x) = f(x+1)

B2 The sequence a1, a2, a3, is defined by a1 = a2 = a3 = 1, an+3 = an+2an+1 + an Show that for

any positive integer r we can find s such that as is a multiple of r

B3 Find the largest possible volume for a tetrahedron which lies inside a hemisphere of

radius 1

A1 An even number of politicians are sitting at a round table After a break, they come back

and sit down again in arbitrary places Show that there must be two people with the same

number of people sitting between them as before the break

A2 k1, k2, k3 are three circles k2 and k3 touch externally at P, k3 and k1 touch externally at Q,

U, V are collinear

A3 The edges of a cube are labeled from 1 to 12 Show that there must exist at least eight

triples (i, j, k) with 1 ≤ i < j < k ≤ 12 so that the edges i, j, k are consecutive edges of a path

But show that the labeling can be done so that we cannot find nine such triples

B1 n, k are positive integers A0 is the set {1, 2, , n} Ai is a randomly chosen subset of Ai-1

is n/2k

B2 Three circles of radius a are drawn on the surface of a sphere of radius r Each pair of

circles touches externally and the three circles all lie in one hemisphere Find the radius of a

circle on the surface of the sphere which touches all three circles

B3 Show that for positive reals a, b, c, d we have ((ab + ac + ad + bc + bd + cd)/6)1/2 ≥ ((abc

+ abd + acd + bcd)/4)1/3

A3 In a tournament there are n players Each pair of players play each other just once There

are no draws Show that either (1) one can divide the players into two groups A and B, such

that every player in A beat every player in B, or (2) we can label the players P1, P2, , Pn such

that Pi beat Pi+1 for i = 1, 2, n (where we use cyclic subscripts, so that Pn+1 means P1)

B1 A triangle with each side length at least 1 lies inside a square side 1 Show that the center

of the square lies inside the triangle

B2 a1, a2, a3, is a sequence of positive integers such that limn→∞ n/an = 0 Show that we can

find k such that there are at least 1990 squares between a1 + a2 + + ak and a1 + a2 + + ak+1

B3 Show that ∑k=0[n/3] (-1)k nC3k is a multiple of 3 for n > 2 (nCm is the binomial

coefficient)

A1 Do there exist tetrahedra T1, T2 such that (1) vol T1 > vol T2, and (2) every face of T2 has

larger area than any face of T1?

A2 Let F(n) be the number of paths P0, P1, , Pn of length n that go from P0 = (0,0) to a

lattice point Pn on the line y = 0, such that each Pi is a lattice point and for each i < n, Pi and

Pi+1 are adjacent lattice points a distance 1 apart Show that F(n) = (2n)Cn

A3 N is a number of the form ∑k=160 ak kkk, where each ak = 1 or -1 Show that N cannot be a

5th power

B1 Let V be the set of all vectors (x,y) with integral coordinates Find all real-valued

functions f on V such that (a) f(v) = 1 for all v of length 1; (b) f(v + w) = f(v) + f(w) for all

B2 k1, k2 are circles with different radii and centers K1, K2 Neither lies inside the other, and

meet at B on K1K2 P is any point on k1 Show that there is a diameter of K2 with one endpoint

on the line PA and the other on the line PB

B3 The real numbers x, y, z satisfy x2 + y2 + z2 = 2 Show that x + y + z ≤ 2 + xyz When do

we have equality?

43rd Polish 1992

A1 Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC| O is the circumcenter of the

triangle PAB Show that OP and CD are perpendicular

A2 Find all functions f : Q+ → Q+, where Q+ is the positive rationals, such that f(x+1) = f(x)

+ 1 and f(x3) = f(x)3 for all x

A3 Show that for real numbers x1, x2, , xn we have ∑i=1m (∑j=1n xixj/(i+j) ) ≥ 0 When do we

have equality?

B1 The functions f0, f1, f2, are defined on the reals by f0(x) = 8 for all x, fn+1(x) = √(x2 +

6fn(x)) For all n solve the equation fn(x) = 2x

B2 The base of a regular pyramid is a regular 2n-gon A1A2 A2n A sphere passes through the

apex S of the pyramid and cuts the edge SAi at Bi (for i = 1, 2, , 2n) Show that ∑ SB2i-1 = ∑

SB2i

B3 Show that k3! is divisible by (k!)k2+k+1

A2 A circle center O is inscribed in the quadrilateral ABCD AB is parallel to and longer

than CD and has midpoint M The line OM meets CD at F CD touches the circle at E Show

that DE = CF iff AB = 2CD

A3 g(k) is the greatest odd divisor of k Put f(k) = k/2 + k/g(k) for k even, and 2(k+1)/2 for k

odd Define the sequence x1, x2, x3, by x1 = 1, xn+1 = f(xn) Find n such that xn = 800

B1 P is a convex polyhedron with all faces triangular The vertices of P are each colored with

one of three colors Show that the number of faces with three vertices of different colors is

even

B2 Find all real-valued functions f on the reals such that f(-x) = -f(x), f(x+1) = f(x) + 1 for

all x, and f(1/x) = f(x)/x2 for x ≠ 0

B3 Is the volume of a tetrahedron determined by the areas of its faces and its circumradius?

45th Polish 1994

A1 Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are

all integers

A2 L, L' are parallel lines C is a circle that does not intersect L A is a variable point on L

The two tangents to C from A meet L' in two points with midpoint M Show that the line AM

passes through a fixed point (as A varies)

A3 k is a fixed positive integer Let an be the number of maps f from the subsets of {1, 2, ,

n} to {1, 2, , k} such that for all subsets A, B of {1, 2, , n} we have f(A ∩ B) = min(f(A),

f(B)) Find limn→∞ an1/n

B1 m, n are relatively prime We have three jugs which contain m, n and m+n liters Initially

the largest jug is full of water Show that for any k in {1, 2, , m+n} we can get exactly k

liters into one of the jugs

B2 A parallelepiped has vertices A1, A2, , A8 and center O Show that 4 ∑ |OAi|2 ≤

(∑|OAi|)2

B3 The distinct reals x1, x2, , xn (n > 3) satisfy ∑ xi = 0, &sum xi2 = 1 Show that four of

the numbers a, b, c, d must satisfy a + b + c + nabc ≤ ∑ xi3 ≤ a + b + d + nabd

A1 How many subsets of {1, 2, , 2n} do not contain two numbers with sum 2n+1?

A2 The diagonals of a convex pentagon divide it into a small pentagon and ten triangles

What is the largest number of the triangles that can have the same area?

A3 p ≥ 5 is prime The sequence a0, a1, a2, is defined by a0 = 1, a1 = 1, , ap-1 = p-1 and an

= an-1 + an-p for n ≥ p Find ap3 mod p

B1 The positive reals x1, x2, , xn have harmonic mean 1 Find the smallest possible value

of x1 + x22/2 + x33/3 + + xnn/n

B2 An urn contains n balls labeled 1, 2, , n We draw the balls out one by one (without

replacing them) until we obtain a ball whose number is divisible by k Find all k such that the

expected number of balls removed is k

B3 PA, PB, PC are three rays in space Show that there is just one pair of points B', C' with

B' on the ray PB and C' on the ray PC such that PC' + B'C' = PA + AB' and PB' + B'C' = PA +

AC'

47th Polish 1996

A1 Find all pairs (n,r) with n a positive integer and r a real such that 2x2+2x+1 divides

(x+1)n - r

A2 P is a point inside the triangle ABC such that ∠PBC = ∠PCA < ∠PAB The line PB

meets the circumcircle of ABC again at E The line CE meets the circumcircle of APE again

at F Show that area APEF/area ABP does not depend on P

A3 ai, xi are positive reals such that a1 + a2 + + an = x1 + x2 + + xn = 1 Show that 2 ∑i<j

xixj ≤ (n-2)/(n-1) + ∑ aixi2/(1-ai) When do we have equality?

B1 ABCD is a tetrahedron with ∠BAC = ∠ACD and ∠ABD = ∠BDC Show that AB =

CD

B2 Let p(k) be the smallest prime not dividing k Put q(k) = 1 if p(k) = 2, or the product of

all primes < p(k) if p(k) > 2 Define the sequence x0, x1, x2, by x0 = 1, xn+1 = xnp(xn)/q(xn)

Find all n such that xn = 111111

B3 Let S be the set of permutations a1a2 an of 123 n such that ai ≥ i An element of S is

≤ i+1 exceeds 1/3

A1 The positive integers x1, x2, , x7 satisfy x6 = 144, xn+3 = xn+2(xn+1+xn) for n = 1, 2, 3, 4

Find x7

A2 Find all real solutions to 3(x2 + y2 + z2) = 1, x2y2 + y2z2 + z2x2 = xyz(x + y + z)3

A3 ABCD is a tetrahedron DE, DF, DG are medians of triangles DBC, DCA, DAB The

angles between DE and BC, between DF and CA, and between DG and AB are equal Show

that area DBC ≤ area DCA + area DAB

B1 The sequence a1, a2, a3, is defined by a1 = 0, an = a[n/2] + (-1)n(n+1)/2 Show that for any

positive integer k we can find n in the range 2k ≤ n < 2k+1 such that an = 0

B2 ABCDE is a convex pentagon such that DC = DE and ∠C = ∠E = 90o F is a point on

B3 Given any n points on a unit circle show that at most n2/3 of the segments joining two

points have length > √2

49th Polish 1998

A1 Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc

A2 Fn is the Fibonacci sequence F0 = F1 = 1, Fn+2 = Fn+1 + Fn Find all pairs m > k ≥ 0 such

that the sequence x0, x1, x2, defined by x0 = Fk/Fm and xn+1 = (2xn - 1)/(1 - xn) for xn ≠ 1, or 1

if xn = 1, contains the number 1

A3 PABCDE is a pyramid with ABCDE a convex pentagon A plane meets the edges PA,

PB, PC, PD, PE in points A', B', C', D', E' distinct from A, B, C, D, E and P For each of the

quadrilaterals ABB'A', BCC'B, CDD'C', DEE'D', EAA'E' take the intersection of the

diagonals Show that the five intersections are coplanar

B1 Define the sequence a1, a2, a3, by a1 = 1, an = an-1 + a[n/2] Does the sequence contain

infinitely many multiples of 7?

B2 The points D, E on the side AB of the triangle ABC are such that (AD/DB)(AE/EB) =

B3 S is a board containing all unit squares in the xy plane whose vertices have integer

square of S An allowed move is to change the sign of every square in S in a given row,

column or diagonal Can we end up with all -1s by a sequence of allowed moves?

A1 D is a point on the side BC of the triangle ABC such that AD > BC E is a point on the

side AC such that AE/EC = BD/(AD-BC) Show that AD > BE

A2 Given 101 distinct non-negative integers less than 5050 show that one choose four a, b,

c, d such that a + b - c - d is a multiple of 5050

A3 Show that one can find 50 distinct positive integers such that the sum of each number

and its digits is the same

B1 For which n do the equations have a solution in integers:

B2 Show that ∑1≤i<j≤n (|ai-aj| + |bi-bj|) ≤ ∑1≤i<j≤n |ai-bj| for all integers ai, bi

B3 The convex hexagon ABCDEF satisfies ∠A + ∠C + ∠E = 360o and AB·CD·EF =

BC·DE·FA Show that AB·FD·EC = BF·DE·CA

A2 The triangle ABC has AC = BC P is a point inside the triangle such that ∠PAB =

A3 The sequence a1, a2, a3, is defined as follows a1 and a2 are primes an is the greatest

B1 PA1A2 An is a pyramid The base A1A2 An is a regular n-gon The apex P is placed so

that the lines PAi all make an angle 60o with the plane of the base For which n is it possible to

find Bi on PAi for i = 2, 3, , n such that A1B2 + B2B3 + B3B4 + + Bn-1Bn + BnA1 < 2A1P?

B2 For each n ≥ 2 find the smallest k such that given any subset S of k squares on an n x n

chessboard we can find a subset T of S such that every row and column of the board has an

even number of squares in T

B3 p(x) is a polynomial of odd degree which satisfies p(x2-1) = p(x)2 - 1 for all x Show that

p(x) = x

A1 Show that x1 + 2x2 + 3x3 + + nxn ≤ ½n(n-1) + x1 + x22 + x33 + + xnn for all

non-negative reals xi

A2 P is a point inside a regular tetrahedron with edge 1 Show that the sum of the distances

from P to the vertices is at most 3

A3 The sequence x1, x2, x3, is defined by x1 = a, x2 = b, xn+2 = xn+1 + xn, where a and b are

reals A number c is a repeated value if it occurs in the sequence more than once Show that

we can choose a, b so that the sequence has more than 2000 repeated values, but not so that it

has infinitely many repeated values

B1 a and b are integers such that 2na + b is a square for all non-negative integers n Show

that a = 0

B2 ABCD is a parallelogram K is a point on the side BC and L is a point on the side CD

B3 Given a set of 2000 distinct positive integers under 10100, show that one can find two

non-empty disjoint subsets which have the same number of elements, the same sum and the

same sum of squares

53rd Polish 2002

A1 Find all triples of positive integers (a, b, c) such that a2 + 1 and b2 + 1 are prime and (a2 +

1)(b2 + 1) = c2 + 1

A2 ABC is an acute-angled triangle BCKL, ACPQ are rectangles on the outside of two of

the sides and have equal area Show that the midpoint of PK lies on the line through C and the

circumcenter

A3 Three non-negative integers are written on a blackboard A move is to replace two of the

integers by their sum and (non-negative) difference Can we always get two zeros by a

sequence of moves?

B1 Given any finite sequence x1, x2, , xn of at least 3 positive integers, show that either

∑1n xi/(xi+1 + xi+2) ≥ n/2 or ∑ 1n xi/(xi-1 + xi-2) ≥ n/2 (We use the cyclic subscript convention, so

that xn+1 means x1 and x-1 means xn-1 etc)

B2 ABC is a triangle A sphere does not intersect the plane of ABC There are 4 points K, L,

M, P on the sphere such that AK, BL, CM are tangent to the sphere and AK/AP = BL/BP =

CM/CP Show that the sphere touches the circumsphere of ABCP

B3 k is a positive integer The sequence a1, a2, a3, is defined by a1 = k+1, an+1 = an - kan +

k Show that am and an are coprime (for m ≠ n)

A1 ABC is acute-angled M is the midpoint of AB A line through M meets the lines CA, CB

at K, L with CK = CL O is the circumcenter of CKL and CD is an altitude of ABC Show that

B1 p is a prime and a, b, c, are distinct positive integers less than p such that a3 = b3 = c3 mod

p Show that a2 + b2 + c2 is divisible by a + b + c

B2 ABCD is a tetrahedron The insphere touches the face ABC at H The exsphere opposite

D (which also touches the face ABC and the three planes containing the other faces) touches

the face ABC at O If O is the circumcenter of ABC, show that H is the orthocenter of ABC

B3 n is even Show that there is a permutation a1a2 an of 12 n such that ai+1 ∈ {2ai, 2ai-1,

means a1)

Spanish (1990 – 2003)

A1 Show that √x + √y + √(xy) = √x + √(y + xy + 2y√x) Hence show that √3 + √(10 + 2√3)

= √(5 + √22) + √(8 - √22 + 2√(15 - 3√22))

A2 Every point of the plane is painted with one of three colors Can we always find two

points a distance 1 apart which are the same color?

A3 Show that [(4 + √11)n] is odd for any positive integer n

B1 Show that ((a+1)/2 + ((a+3)/6)√((4a+3)/3) )1/3 + ((a+1)/2 - ((a+3)/6)√((4a+3)/3) )1/3 is

independent of a for a ≥ 3/4 and find it

B2 ABC is a triangle with area S Points A', B', C' are taken on the sides BC, CA, AB, so

that AC'/AB = BA'/BC = CB'/CA = k, where 0 < k < 1 Find the area of A'B'C' in terms of S

and k Find the value of k which minimises the area The line through A' parallel to AB and

the line through C' parallel to AC meet at P Find the locus of P as k varies

B3 There are n points in the plane so that no two pairs are the same distance apart Each

point is connected to the nearest point by a line Show that no point is connected to more than

5 points

27th Spanish 1991

A1 Let S be the set of all points in the plane with integer coordinates Let T be the set of all

A2 a, b are distinct elements of {0,1,-1} A is the matrix:

a+b a+b2 a+b3 a+bm

a2+b a2+b2 a2+b3 a2+bm

a3+b a3+b2 a3+b3 a3+bm

an+b an+b2 an+b3 an+bm

Find the smallest possible number of columns of A such that any other column is a linear

combination of these columns with integer coefficients

A3 What condition must be satisfied by the coefficients u, v, w if the roots of the polynomial

x3 - ux2 + vx - w can be the sides of a triangle?

B1 The incircle of ABC touches BC, CA, AB at A', B', C' respectively The line A'C' meets

B2 Let s(n) be the sum of the binary digits of n Find s(1) + s(2) + s(3) + + s(2k) for each

positive integer k

B3 Find the integral part of 1/√1 + 1/√2 + 1/√3 + + 1/√1000

A1 Find the smallest positive integer N which is a multiple of 83 and is such that N2 has

exactly 63 positive divisors

A2 Given two circles (neither inside the other) with different radii, a line L, and k > 0, show

how to construct a line L' parallel to L so that L intersects the two circles in chords with total

length k

A3 a, b, c, d are positive integers such that (a+b)2 + 2a + b = (c+d)2 + 2c + d Show that a = c

and b = d Show that the same is true if a, b, c, d satisfy (a+b)2 + 3a + b = (c+d)2 + 3c + d But

show that there exist a, b, c, d such that (a+b)2 + 4a + b = (c+d)2 + 4c + d, but a ≠ c and b ≠ d

B1 Show that there are infinitely many primes in the arithmetic progression 3, 7, 11, 15,

B2 Given the triangle ABC, show how to find geometrically the point P such that ∠PAB =

functions

B3 For each positive integer n let S(n) be the set of complex numbers z such that |z| = 1 and

(z + 1/z)n = 2n-1(zn + 1/zn) Find S(2), S(3), S(4) Find an upper bound for |S(n)| for n ≥ 5

29th Spanish 1993

A1 There is a reunion of 201 people from 5 different countries In each group of 6 people, at

least two have the same age Show that there must be at least 5 people with the same country,

age and sex

A2 In the triangle of numbers below, each number is the sum of the two immediately above:

0 1 2 3 4 1991 1992 1993

1 3 5 7 3983 3985

4 8 12 7968

Show that the bottom number is a multiple of 1993

A3 Show that for any triangle 2r ≤ R (where r is the inradius and R is the circumradius)

B1 Show that for any prime p ≠ 2, 5, infinitely many numbers of the form 11 1 are

multiples of p

B2 Given a 4 x 4 grid of points as shown below The points at two opposite corners are

marked A and D as shown How many ways can we choose a set of two further points {B,C}

so that the six distances between A, B, C, D are all distinct?

How many of the sets of 4 points are geometrically distinct (so that one cannot be obtained

from another by a reflection, rotation etc)? Give the points coordinates (x,y) from (1,1) to

(4,4) Take the grid-distance between (x,y) and (u,v) to be |x-u| + |y-v| Show that the sum of

the six grid-distances between the points is always the same

B3 A casino game uses the diagram shown At the start a ball appears at S Each time the

player presses a button, the ball moves to one of the adjacent letters (joined by a line segment)

(with equal probability) If the ball returns to S the player loses If the ball reaches G, then the

player wins Find the probability that the player wins and the expected number of button

presses

A1 Show that if an (infinite) arithmetic progression includes a square, then it must include

infinitely many squares

A2 Take three-dimensional coordinates with origin O C is the point (0,0,c) P is a point on

the x-axis, and Q is a point on the y-axis such that OP + OQ = k, where k is fixed Let W be

the center of the sphere through O, C, P, Q Let W' be the projection of W on the xy-plane

Find the locus of W' as P and Q vary Find also the locus of W as P and Q vary

A3 The tourism office is collecting figures on the number of sunny days and the number of

rainy days in the regions A, B, C, D, E, F

If one region is excluded then the total number of rainy days in the other regions is one-third

of the total number of sunny days in those regions Which region is excluded?

B1 The triangle ABC has ∠A = 36o, ∠B = 72o, ∠C = 72o The bisector of ∠C meets AB

at D Find the angles of BCD Express the length BC in terms of AC, without using any

trigonometric functions

B2 21 counters are arranged in a 3 x 7 grid Some of the counters are black and some white

Show that one can always find 4 counters of the same color at the vertices of a rectangle

B3 A convex n-gon is divided into m triangles, so that no two triangles have interior points

in common, and each side of a triangle is either a side of the polygon or a side of another

triangle Show that m + n must be even Given m, n, find the number of triangle sides in the

interior of the polygon and the number of vertices in the interior of the polygon

31st Spanish 1995

A1 X is a set of 100 distinct positive integers such that if a, b, c ∈ X (not necessarily

distinct), then there is a triangle with sides a, b, c which is not obtuse Let S(X) be the sum of

the perimeters of all the possible triangles Find the minimum possible value of S(X)

A2 A finite number of paper disks are arranged so that no disk lies inside another, but there

is some overlapping Show that if we cut out the parts which do not overlap we cannot

rearrange them to form disks

A3 ABC is a triangle with centroid G A line through G meets the side AB at P and the side

AC at Q Show that (PB/PA)(QC/QA) ≤ 1/4

B1 p is a prime number Find all integral solutions to p(m+n) = mn

B2 Given that the equations x3 + mx - n = 0, nx3 - 2m2x2 - 5mnx - 2m3 - n2 = 0 (where n ≠ 0)

have a common root, show that the first must have two equal roots and find the roots of the

two equations in terms of n

B3 C is a variable point on the segment AB Equilateral triangles AB'C and BA'C are

constructed on the same side of AB, and the equilateral triangle ABC' is constructed on the

opposite side of AB Show that AA', BB', CC' meet at some point P Find the locus of P as C

varies Show that the centers of the three equilateral triangles form an equilateral triangle and

lie on a fixed circle (as C varies)

A1 The integers m, n are such that (m+1)/n + (n+1)/m is an integer Show that gcd(m,n) ≤

B1 Discuss the existence of solutions to the equation √(x2-p) + 2√(x2-1) = x for varying

values of the real parameter p

B2 In Port Aventura there are 16 secret agents Each of the agents watches some of his

rivals It is known that if agent A watches agent B, then agent B does not watch agent A It is

possible to find 10 agents such that the first watches the second, the second watches the third,

, and the tenth watches the first Show that it is possible to find a cycle of 11 such agents

B3 Take a cup made of 6 regular pentagons, so that two such cups could be put together to

form a regular dodecahedron The edge length is 1 What volume of liquid will the cup hold?

33rd Spanish 1997

A1 An arithmetic progression has 100 terms The sum of the terms is -1, and the sum of the

even-numbered terms is 1 Find the sum of the squares of the terms

A2 X is the set of 16 points shown What is the largest number of elements of X that we can

choose so that no three of the chosen points form an isosceles triangle?

A3 Let S be the set of all parabolas y = x2 + px + q whose graphs cut the coordinate axes in

three distinct points Let C(p,q) be the circle through the three points Show that all circles

C(p,q) have a common point

B1 Given a prime p, find all integers k such that √(k2-kp) is integral

B2 Q is a convex quadrilateral with area 1 Show that the sum of the sides and diagonals is

at least 2(2+√2)

B3 A car wishes to make a circuit of a circular road There are some tanks along the road

that contain between them just enough gasoline for the car to make the trip The car has a tank

large enough to hold all the gasoline for a complete circuit, but the tank is initially empty

Show that irrespective of the number of tanks, their positions, and the amount each contains,

it is possible to find a starting point on the road which will allow the car to make a complete

circuit (refueling when it reaches a tank)

A1 A square side 1 is rotated through an angle θ about its center Find the area common to

the original and rotated squares

A2 Find all 4-digit numbers which equal the cube of the sum of their digits

A3 ABC is a triangle D, E are points on the line BC such that AD, AE are parallel to the

B1 A triangle has angles A, B, C such that tan A, tan B, tan C are all positive integers Find

A, B, C

B2 Let N be the set of positive integers Find all functions f : N → N which are strictly

increasing and which satisfy f(n + f(n)) = 2 f(n) for all n

B3 For which values of n is it possible to tile an n x n square with tiles of the type:

35th Spanish 1999

A1 A and B are points of the parabola y = x2 The tangents at A and B meet at C The

median of the triangle ABC from C has length m Find area ABC in terms of m

A2 Show that there is an infinite sequence a1, a2, a3, of positive integers such that a12 + a22

+ + an2 is a square for all n

A3 A game is played on the board shown A token is placed on each circle Each token has a

black side and a white side Initially the topmost token has the black face showing, the others

have the white face showing A move is to remove token showing its black face and to turn

over the tokens on the adjacent circles (joined by a line) Is it possible to remove all the

tokens by a sequence of moves?

B1 A box contains 900 cards, labeled from 100 to 999 Cards are removed one at a time

without replacement What is the smallest number of cards that must be removed to guarantee

that at least three of the digit sums of the cards removed are equal?

B2 G is the centroid of the triangle ABC The distances of G from the three sides are ga, gb,

gc Show that ga ≥ 2r/3, and (ga + gb + gc) ≥ 3r, where r is the inradius

B3 Three families of parallel lines divide the plane into N regions No three lines pass

through the same point What is the smallest number of lines needed to get N > 1999?

A1 Let p(x) = x4 + ax3 + bx2 + cx + 1, q(x) = x4 + cx3 + bx2 + ax + 1 Find conditions on a, b,

c (assuming a ≠ c) so that p(x) and q(x) have two common roots In this case solve p(x) = q(x)

= 0

A2 The diagram shows a network of roads The distance from one node to an adjacent node

is 1 P goes from A to B by a path length 7, and Q goes from B to A by a path length 7 Each

goes at the same constant speed At each junction with two possible directions to take, each

has probability 1/2 Find the probability that P and Q meet

A3 Circles C and C' meet at A and B P, P' are variable points on C, C' such that P, B, P' are

collinear Show that the perpendicular bisector of PP' passes through a fixed point M (which

depends only on C and C')

B1 Find the largest integer N such that [N/3] has three digits, all equal, and [N/3] = 1 + 2 + 3