Đề olympic toán quốc tế international tại trung quốc

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Đề olympic toán quốc tế international tại trung quốc

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A collection of IMO Team Selection Tests September 22, 2013 Contents Preface China Team Selection Tests Germany Team Selection Tests 46 India Training Camp Tests 72 Iran Pre-preparation Course Examination 92 Iran Team Selection Tests 106 Italy Team Selection Tests 121 Romanian Team Selection Tests 129 Turkey Team Selection Tests 168 USA Team Selection Tests 186 USA ELMO Tests 198 USA ELMO Shorlists 203 Vietnam Team Selection Tests 217 Glossary 237 Preface Due to problems with the pdf generator at AoPS I decided to make my own collection of math Olympiad problems 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 a new 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 The links 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 digits, but earlier posts may have 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 and there are constantly new contests being written Therefore I start with the most popular contest, and the ones I have most complete collections of Also, in stead of defining common terms over and over in problems that refer to them, 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 the appearance 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 superfluous definitions like “where R is the set of real numbers” or “where x denotes the greatest integer ”, etc China Team Selection Tests China Team Selection Test 1986 If ABCD is a cyclic quadrilateral, then prove that the incentres of the triangles ABC, BCD, CDA, DAB are the vertices of a rectangle AoPS:154713 Let a1 , a2 , , an and b1 , b2 , , bn be · n real numbers Prove that the following two statements are equivalent: (a) For any n real numbers x1 , x2 , , xn satisfying x1 ≤ x2 ≤ ≤ xn , we have n k=1 bk · xk ; s k=1 (b) We have ak ≤ s k=1 bk for every s ∈ {1, 2, , n − 1} and n k=1 ak = n k=1 ak · xk ≤ n k=1 bk AoPS:234015 Given a positive integer A written in decimal expansion: (an , an−1 , , a0 ) and let f (A) denote n n−k · ak Define A1 = f (A), A2 = f (A1 ) Prove that: k=0 (a) There exists positive integer k for which Ak+1 = Ak (b) Find such Ak for 1986 AoPS:234060 Given a triangle ABC for which C = 90 degrees, prove that given n points inside it, we can name them P1 , P2 , , Pn in some way such that: n−1 k=1 (PK Pk+1 ) ≤ AB (the sum is over the consecutive square of the segments from up to n − 1) AoPS:234065 Given a square ABCD whose side length is 1, P and Q are points on the sides AB and AD If the perimeter of AP Q is find the angle P CQ AoPS:234067 Given a tetrahedron ABCD, E, F , G, are on the respectively on the segments AB, AC and AD Prove that: (a) area EF G ≤ max(area ABC, area ABD, area ACD, area BCD) (b) The same as above replacing “area” with “perimeter” AoPS:234069 Let xi , ≤ i ≤ n be real numbers with n ≥ Let p and q be their symmetric sum of degree and respectively Prove that: (a) p2 · n−1 n (b) xi − p n − 2q ≥ 0; ≤ p2 − 2nq n−1 · n−1 n for every meaningful i AoPS:234075 Mark · k points in a circle and number them arbitrarily with numbers from to · k The chords cannot share common endpoints, also, the endpoints of these chords should be among the · k points (a) Prove that · k pairwise non-intersecting chords can be drawn for each of whom its endpoints differ in at most · k − (b) Prove that the · k − cannot be improved AoPS:234076 China Team Selection Test 1987 For all positive integer k find the smallest positive integer f (k) such that sets s1 , s2 , , s5 exist satisfying: (a) i each has k elements; ii si and si+1 are disjoint for i = 1, 2, , (s6 = s1 ) iii the union of the sets has exactly f (k) elements (b) Generalisation: Consider n ≥ sets instead of AoPS:234080 A closed polygon with 100 sides (may be concave) is given such that its vertices have integer coordinates, its sides are parallel to the axis and all its sides have odd length Prove that its area is odd AoPS:234083 n−1 Let r1 = and rn = k=1 ri + 1, n ≥ Prove that among all sets of positive integers such that n k=1 < 1, the partial sequences r1 , r2 , , rn are the one that gets nearer to AoPS:234089 Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle A inside it with maximum area (over all possible rectangles) Then we enlarge it with centre in the centre of the rectangle and ratio λ such that is covers the convex figure Find the smallest λ such that it works for all convex figures AoPS:234092 Find all positive integer n such that the equation x3 + y + z = n · x2 · y · z has positive integer solutions AoPS:234095 Let G be a simple graph with · n vertices and n2 + edges Show that this graph G contains a K4 -one edge, that is, two triangles with a common edge AoPS:234099 China Team Selection Test 1988 Suppose real numbers A, B, C such that for all real numbers x, y, z the following inequality holds: A(x − y)(x − z) + B(y − z)(y − x) + C(z − x)(z − y) ≥ Find the necessary and sufficient condition A, B, C must satisfy (expressed by means of an equality or an inequality) AoPS:269050 Find all functions f : Q → C satisfying (a) For any x1 , x2 , , x1988 ∈ Q, f (x1 + x2 + + x1988 ) = f (x1 )f (x2 ) f (x1988 ) (b) f (1988)f (x) = f (1988)f (x) for all x ∈ Q AoPS:269052 In triangle ABC, ∠C = 30◦ , O and I are the circumcentre and incentre respectively, Points D ∈ AC and E ∈ BC, such that AD = BE = AB Prove that OI = DE and OI⊥DE AoPS:269054 Let k ∈ N, Sk = {(a, b)|a, b = 1, 2, , k} Any two elements (a, b), (c, d) ∈ Sk are called undistinguishing in Sk if a − c ≡ or ±1 (mod k) and b − d ≡ or ±1 (mod k); otherwise, we call them distinguishing For example, (1, 1) and (2, 5) are undistinguishing in S5 Considering the subset A of Sk such that the elements of A are pairwise distinguishing Let rk be the maximum possible number of elements of A (a) Find r5 (b) Find r7 (c) Find rk for k ∈ N AoPS:269055 Let f (x) = 3x + Prove that there exists m ∈ N such that f 100 (m) is divisible by 1988 AoPS:269053 Let ABCD be a trapezium AB//CD, M and N are fixed points on AB, P is a variable point on CD E = DN ∩ AP , F = DN ∩ M C, G = M C ∩ P B, DP = λ · CD Find the value of λ for which the area of quadrilateral P EF G is maximum AoPS:269056 A polygon is given in the OXY plane and its area exceeds n Prove that there exist n + points P1 (x1 , y1 ), P2 (x2 , y2 ), , Pn+1 (xn+1 , yn+1 ) in such that ∀i, j ∈ {1, 2, , n + 1}, xj − xi and yj − yi are all integers AoPS:269057 There is a broken computer such that only three primitive data c, and −1 are reserved Only allowed operation may take u and v and output u · v + v At the beginning, u, v ∈ {c, 1, −1} After then, it can also take the value of the previous step (only one step back) besides {c, 1, −1} Prove that for any polynomial Pn (x) = a0 · xn + a1 · xn−1 + + an with integer coefficients, the value of Pn (c) can be computed using this computer after only finite operation AoPS:269058 China Team Selection Test 1989 √ √ A triangle of sides 23 , 25 , is folded along a variable line perpendicular to the side of 32 Find the maximum value of the coincident area AoPS:269059 Let v0 = 0, v1 = and vn+1 = · − vn−1 , n = 1, 2, Prove that in the sequence {vn } there is no term of the form 3α · 5β with α, β ∈ N AoPS:269060 Find the greatest n such that (z+1)n = z n +1 has all its non-zero roots in the unitary circumference, e.g (α + 1)n = αn + 1, α = implies |α| = AoPS:269061 Given triangle ABC, squares ABEF, BCGH, CAIJ are constructed externally on side AB, BC, CA, respectively Let AH ∩ BJ = P1 , BJ ∩ CF = Q1 , CF ∩ AH = R1 , AG ∩ CE = P2 , BI ∩ AG = Q2 , CE ∩ BI = R2 Prove that triangle P1 Q1 R1 is congruent to triangle P2 Q2 R2 AoPS:269062 Let N = {1, 2, } Does there exists a function f : N → N such that ∀n ∈ N, f 1989 (n) = · n ? AoPS:269063 AD is the altitude on side BC of triangle ABC If BC + AD − AB − AC = 0, find the range of ∠BAC AoPS:269064 1989 equal circles are arbitrarily placed on the table without overlap What is the least number of colours are needed such that all the circles can be painted with any two tangential circles coloured differently AoPS:269065 ∀n ∈ N, P (n) denotes the number of the partition of n as the sum of positive integers (disregarding the order of the parts), e.g since = + + + = + + = + = + = 4, so P (4) = The dispersion of a partition denotes the number of different parts in that partition And denote q(n) is the sum of all the dispersions, e.g q(4) = + + + + = n ≥ Prove that (a) q(n) = + (b) + n−1 i=1 n−1 i=1 P (i) √ P (i) ≤ · n · P (n) AoPS:269066 China Team Selection Test 1990 In a wagon, every m ≥ people have exactly one common friend (When A is B’s friend, B is also A’s friend No one was considered as his own friend.) Find the number of friends of the person who has the most friends AoPS:269067 Finitely many polygons are placed in the plane If for any two polygons of them, there exists a line through origin O that cuts them both, then these polygons are called properly placed Find the least m ∈ N, such that for any group of properly placed polygons, m lines can drawn through O and every polygon is cut by at least one of these m lines AoPS:269068 In set S, there is an operation ◦ such that ∀a, b ∈ S, a unique a ◦ b ∈ S exists And (i) ∀a, b, c ∈ S, (a ◦ b) ◦ c = a ◦ (b ◦ c) (ii) a ◦ b = b ◦ a when a = b Prove that: (a) ∀a, b, c ∈ S, (a ◦ b) ◦ c = a ◦ c (b) If S = {1, 2, , 1990}, try to define an operation ◦ in S with the above properties AoPS:269069 Number a is such that ∀a1 , a2 , a3 , a4 ∈ R, there are integers k1 , k2 , k3 , k4 such that ki ) − (aj − kj ))2 ≤ a Find the minimum of a 1≤i 0, i = 1, 2, , n, we have k Sk = k where Sk = i=1 i=1 AoPS:269091 China Team Selection Test 1993 For all primes p ≥ 3, define F (p) = p−1 k=1 k 120 and f (p) = − F (p) p , where {x} = x − [x], find the value of f (p) AoPS:269092 Let n ≥ 2, n ∈ N, a, b, c, d ∈ N, n a b + dc < and a + c ≤ n, find the maximum value of a b + dc for fixed AoPS:269096 A graph G = (V, E) is given If at least n colours are required to paints its vertices so that between any two same coloured vertices no edge is connected, then call this graph n-coloured Prove that for any n ∈ N, there is a n-coloured graph without triangles AoPS:269093 Find all integer solutions to 2x4 + = y AoPS:269097 Let S = {(x, y)|x = 1, 2, , 1993, y = 1, 2, 3, 4} If T ⊂ S and there are no squares in T Find the maximum possible value of |T | The squares in T use points in S as vertices AoPS:269098 Let ABC be a triangle and its bisector at A cuts its circumcircle at D Let I be the incentre of triangle ABC, M be the midpoint of BC, P is the symmetric to I with respect to M (Assuming P is in the circumcircle) Extend DP until it cuts the circumcircle again at N Prove that among segments AN, BN, CN , there is a segment that is the sum of the other two AoPS:269099 China Team Selection Test 1994 Find all sets comprising of natural numbers such that the product of any numbers in the set leaves a remainder of when divided by the remaining number AoPS:234853 An n by n grid, where every square contains a number, is called an n-code if the numbers in every row and column form an arithmetic progression If it is sufficient to know the numbers in certain squares of an n-code to obtain the numbers in the entire grid, call these squares a key (a) Find the smallest s ∈ N such that any s squares in an n−code (n ≥ 4) form a key (b) Find the smallest t ∈ N such that any t squares along the diagonals of an n-code (n ≥ 4) form a key AoPS:234855 Find the smallest n ∈ N such that if any vertices of a regular n-gon are coloured red, there exists a line of symmetry l of the n-gon such that every red point is reflected across l to a non-red point AoPS:234857 n 1 i=1 ri , S = n n n ri si ti ui vi +1 RST U V +1 i=1 ri si ti ui vi −1 ≥ RST U V −1 Given 5n real numbers ri , si , ti , ui , vi ≥ 1(1 ≤ i ≤ n), let R = n n i=1 ti , U= n n i=1 ui , V = n n i=1 vi Prove that n i=1 si , n T = AoPS:234859 Given distinct prime numbers p and q and a natural number n ≥ 3, find all a ∈ Z such that the polynomial f (x) = xn + axn−1 + pq can be factored into integral polynomials of degree at least AoPS:234861 For any convex polygons S and T , if all the vertices of S are vertices of T , call S a sub-polygon of T (a) Prove that for an odd number n ≥ 5, there exists m sub-polygons of a convex n-gon such that they not share any edges, and every edge and diagonal of the n-gon are edges of the m sub-polygons (b) Find the smallest possible value of m AoPS:234862 China Team Selection Test 1995 Find the smallest prime number p that cannot be represented in the form |3a − 2b |, where a and b are non-negative integers AoPS:234864 Given a fixed acute angle θ and a pair of internally tangent circles, let the line l which passes through the point of tangency, A, cut the larger circle again at B (l does not pass through the centres of the circles) Let M be a point on the major arc AB of the larger circle, N the point where AM intersects the smaller circle, and P the point on ray M B such that ∠M P N = θ Find the locus of P as M moves on major arc AB of the larger circle AoPS:234865

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  • Preface

  • China Team Selection Tests

  • Germany Team Selection Tests

  • India Training Camp Tests

  • Iran Pre-preparation Course Examination

  • Iran Team Selection Tests

  • Italy Team Selection Tests

  • Romanian Team Selection Tests

  • Turkey Team Selection Tests

  • USA Team Selection Tests

  • USA ELMO Tests

  • USA ELMO Shorlists

  • Vietnam Team Selection Tests

  • Glossary

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