Tuyển Tập Các Đề Thi Của Các Nước Trên Thế Giới P2

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  1. TRƯỜNG ................ KHOA……… …………..o0o………….. Tuyển tập các đề thi của các nước trên thế giới P2 - Cao Minh Quang
  2. ☺ The best problems from around the world Cao Minh Quang 13th Mexican 1999 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 exterior bisectors of ∠ A and ∠ D meet at Q. Show that PQ is half the perimeter of ABCD. 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. 301
  3. ☺ The best problems from around the world Cao Minh Quang 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 the triangle. 1 2 3 4 5 3 5 7 9 8 12 16 20 28 48 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: a1 = 5, an+1 = a an + b. What is the largest number of primes that can be obtained before the 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. 302
  4. ☺ The best problems from around the world Cao Minh Quang 15th Mexican 2001 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 = AB. Let L be the line through B parallel to AC. Let L meet the external bisector of ∠ A at M 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? 303
  5. ☺ The best problems from around the world Cao Minh Quang 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 = 90o. K is the foot of the perpendicular from M to CD. AK meets BD at P, and BK meets AC at Q. Show that ∠ AKB = 90o and KP/PA + KQ/QB = 1. 304
  6. ☺ The best problems from around the world Cao Minh Quang 17th Mexican 2003 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 K2 is the circle with diameter BC. Another circle touches AC at B and meets K1 again at P and K2 again at Q. The line PQ meets K1 again at R and K2 again at S. Show that the lines AR 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 example, we can form 5 → 11 → 35 so 5, 11 and 35 belong to S5. We call m and n compatible if Sm ∩ Sn is non-empty. Which members of {1, 2, 3, ... , 2002} are compatible with 2003? 305
  7. ☺ The best problems from around the world Cao Minh Quang Polish (1983 – 2003) 306
  8. ☺ The best problems from around the world Cao Minh Quang 34th Polish 1983 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 below) X X . to . . X, or . to X X . X . Show that if mn is a multiple of 3, then it is not possible to end up with only one piece after a sequence of moves. B3. Show that if the positive integers a, b, c, d satisfy ab = cd, then we have gcd(a,c) gcd(a,d) = a gcd(a,b,c,d). 307
  9. ☺ The best problems from around the world Cao Minh Quang 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. 308
  10. ☺ The best problems from around the world Cao Minh Quang 36th Polish 1985 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 ≥ 100. Show that the square can be covered with rectangles Ri with sides length (ai, bi) 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. 309
  11. ☺ The best problems from around the world Cao Minh Quang 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 that there are at most n2 matches. Conversely, show that if there are at most n2 matches, then it 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. 310
  12. ☺ The best problems from around the world Cao Minh Quang 38th Polish 1987 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, Q are distinct vertices of the n-gon. 5 elements L1, L2, ... , L5 of X are chosen at random (so 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. 311
  13. ☺ The best problems from around the world Cao Minh Quang 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. 312
  14. ☺ The best problems from around the world Cao Minh Quang 40th Polish 1989 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, and k1 and k2 touch externally at R. The line PQ meets k1 again at S, the line PR meets k1 again at T. The line RS meets k2 again at U, and the line QT meets k3 again at V. Show that P, 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 (with each subset having equal probability). Show that the expected number of elements of Ak 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. 313
  15. ☺ The best problems from around the world Cao Minh Quang 41st Polish 1990 A1. Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x2- y2) for all x, y. A2. For n > 1 and positive reals x1, x2, ... , xn, show that x12/(x12+x2x3) + x22/(x22+x3x4) + ... + xn2/(xn2+x1x2) ≤ n-1. 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). 314
  16. ☺ The best problems from around the world Cao Minh Quang 42nd Polish 1991 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 perpendicular v, w ∈ V. (The vector (0,0) is considered to be perpendicular to any vector.) B2. k1, k2 are circles with different radii and centers K1, K2. Neither lies inside the other, and they do not touch or intersect. One pair of common tangents meet at A on K1K2, the other pair 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? 315
  17. ☺ The best problems from around the world Cao Minh Quang 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. 316
  18. ☺ The best problems from around the world Cao Minh Quang 44th Polish 1993 A1. Find all rational solutions to: t2 - w2 + z2 = 2xy t2 - y2 + w2 = 2xz t2 - w2 + x2 = 2yz. 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? 317
  19. ☺ The best problems from around the world Cao Minh Quang 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. 318
  20. ☺ The best problems from around the world Cao Minh Quang 46th Polish 1995 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'. 319


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