Vietnam Team Selection Tests 1999 Day 1 1 Let an odd prime p be a given number satisfying 2 h = 1 (mod p) for all h < p − 1, h ∈ N ∗ , and an even integer a ∈  p 2 , p  . Let us consider the sequence {a n } ∞ n=0 defined by a 0 = a and a n+1 = p − b n for n = 0, 1, 2, . . ., where b n is the greatest odd divisor of a n . Show that {a n } is periodical and find its least positive period. 2 Two polynomials f (x) and g(x) with real coefficients are called similar if there exist nonzero real number a such that f (x) = q · g(x) for all x ∈ R. I. Show that there exists a polynomial P (x) of degree 1999 with real coefficients which satisfies the condition: (P (x)) 2 − 4 and (P  (x)) 2 · (x 2 − 4) are similar. II. How many polynomials of degree 1999 are there which have above mentioned property. 3 Let a convex polygon H be given. Show that for every real number a ∈ (0, 1) there ex- ist 6 distinct points on the sides of H, denoted by A 1 , A 2 , . . . , A 6 clockwise, satisfying the conditions: I. (A 1 A 2 ) = (A 5 A 4 ) = a · (A 6 A 3 ). II. Lines A 1 A 2 , A 5 A 4 are equidistant from A 6 A 3 . (By (AB) we denote vector AB) http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 1999 Day 2 1 Let a sequence of positive reals {u n } ∞ n=1 be given. For every positive integer n, let k n be the least positive integer satisfying: k n  i=1 1 i ≥ n  i=1 u i . Show that the sequence  k n+1 k n  has finite limit if and only if {u n } does. 2 Let a triangle ABC inscribed in circle Γ be given. Circle Θ lies in angle of triangle and touches sides AB, AC at M 1 , N 1 and touches internally Γ at P 1 . The points M 2 , N 2 , P 2 and M 3 , N 3 , P 3 are defined similarly to angles B and C respectively. Show that M 1 N 1 , M 2 N 2 and M 3 N 3 intersect each other at their midpoints. 3 Let a regular polygon with p vertices be given, where p is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes p peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex . after giving the k-th peanut, he skips the 2 · k next vertices and gives k + 1-th for the monkey at the next vertex. He does so until all p peanuts are delivered. I. How many monkeys are there which does not receive peanuts? II. How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)? http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2000 Day 1 1 Two circles C 1 and C 2 intersect at points P and Q. Their common tangent, closer to P than to Q, touches C 1 at A and C 2 at B. The tangents to C 1 and C 2 at P meet the other circle at points E = P and F = P , respectively. Let H and K be the points on the rays AF and BE respectively such that AH = AP and BK = BP . Prove that A, H, Q, K, B lie on a circle. 2 Let k be a given positive integer. Dene x 1 = 1 and, for each n > 1, set x n+1 to be the smallest positive integer not belonging to the set {x i , x i + ik|i = 1, ., n}. Prove that there is a real number a such that x n = [an] for all n ∈ N. 3 Two players alternately replace the stars in the expression ∗x 2000 + ∗x 1999 + . + ∗x + 1 by real numbers. The player who makes the last move loses if the resulting polynomial has a real root t with |t| < 1, and wins otherwise. Give a winning strategy for one of the players. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2000 Day 2 1 Let a, b, c be pairwise coprime natural numbers. A positive integer n is said to be stubborn if it cannot be written in the form n = bcx + cay + abz, for some x, y, z ∈ N. Determine the number of stubborn numbers. 2 Let a > 1 and r > 1 be real numbers. (a) Prove that if f : R + → R + is a function satisfying the conditions (i) f(x) 2 ≤ ax r f( x a ) for all x > 0, (ii) f(x) < 2 2000 for all x < 1 2 2000 , then f(x) ≤ x r a 1−r for all x > 0. (b) Construct a function f : R + → R + satisfying condition (i) such that for all x > 0, f(x) > x r a 1−r . 3 A collection of 2000 congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let N be the number of points that belong to at least two of the circles. Find the smallest possible value of N. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2001 Day 1 1 Let a sequence of integers {a n }, n ∈ N be given, defined by a 0 = 1, a n = a n−1 + a [n/3] for all n ∈ N ∗ . Show that for all primes p ≤ 13, there are infinitely many integer numbers k such that a k is divided by p. (Here [x] denotes the integral part of real number x). 2 In the plane let two circles be given which intersect at two points A, B; Let P T be one of the two common tangent line of these circles (P, T are points of tangency). Tangents at P and T of the circumcircle of triangle AP T meet each other at S. Let H be a point symmetric to B under P T . Show that A, S, H are collinear. 3 Some club has 42 members. Its known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2001 Day 2 1 Lets consider the real numbers a, b, c satisfying the condition 21 · a · b + 2 · b · c + 8 · c · a ≤ 12. Find the minimal value of the expression P (a, b, c) = 1 a + 1 b + 1 c . 2 Let an integer n > 1 be given. In the space with orthogonal coordinate system Oxyz we denote by T the set of all points (x, y, z) with x, y, z are integers, satisfying the condition: 1 ≤ x, y, z ≤ n. We paint all the points of T in such a way that: if the point A(x 0 , y 0 , z 0 ) is painted then points B(x 1 , y 1 , z 1 ) for which x 1 ≤ x 0 , y 1 ≤ y 0 and z 1 ≤ z 0 could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied. 3 Let a sequence {a n }, n ∈ N ∗ given, satisfying the condition 0 < a n+1 − a n ≤ 2001 for all n ∈ N ∗ Show that there are infinitely many pairs of positive integers (p, q) such that p < q and a p is divisor of a q . http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2002 Day 1 1 Find all triangles ABC for which ∠ACB is acute and the interior angle bisector of BC intersects the trisectors (AX, (AY of the angle ∠BAC in the points N, P respectively, such that AB = NP = 2DM , where D is the foot of the altitude from A on BC and M is the midpoint of the side BC. 2 On a blackboard a positive integer n 0 is written. Two players, A and B are playing a game, which respects the following rules: − acting alternatively per turn, each player deletes the number written on the blackboard n k and writes instead one number denoted with n k+1 from the set  n k − 1,  n k 3  ; − player A starts first deleting n 0 and replacing it with n 1 ∈  n 0 − 1,  n 0 3  ; − the game ends when the number on the table is 0 - and the player who wrote it is the winner. Find which player has a winning strategy in each of the following cases: a) n 0 = 120; b) n 0 = 3 2002 − 1 2 ; c) n 0 = 3 2002 + 1 2 . 3 Let m be a given positive integer which has a prime divisor greater than √ 2m + 1. Find the minimal positive integer n such that there exists a finite set S of distinct positive integers satisfying the following two conditions: I. m ≤ x ≤ n for all x ∈ S; II. the product of all elements in S is the square of an integer. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2002 Day 2 1 Let n ≥ 2 be an integer and consider an array composed of n rows and 2n columns. Half of the elements in the array are colored in red. Prove that for each integer k, 1 < k ≤  n 2  + 1, there exist k rows such that the array of size k × 2n formed with these k rows has at least k!(n − 2k + 2) (n − k + 1)(n − k + 2)··· (n − 1) columns which contain only red cells. 2 Find all polynomials P (x) with integer coefficients such that the polynomial Q(x) = (x 2 + 6x + 10) · P 2 (x) − 1 is the square of a polynomial with integer coefficients. 3 Prove that there exists an integer n, n ≥ 2002, and n distinct positive integers a 1 , a 2 , . . . , a n such that the number N = a 2 1 a 2 2 ··· a 2 n − 4(a 2 1 + a 2 2 + ··· + a 2 n ) is a perfect square. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2003 Day 1 1 Let be four positive integers m, n, p, q, with p < m given and q < n. Take four points A(0; 0), B(p; 0), C(m; q) and D(m; n) in the coordinate plane. Consider the paths f from A to D and the paths g from B to C such that when going along f or g, one goes only in the positive directions of coordinates and one can only change directions (from the positive direction of one axe coordinate into the the positive direction of the other axe coordinate) at the points with integral coordinates. Let S be the number of couples (f, g) such that f and g have no common points. Prove that S =  n m + n  ·  q m + q − p  −  q m + q  ·  n m + n − p  . 2 Given a triangle ABC. Let O be the circumcenter of this triangle ABC. Let H, K, L be the feet of the altitudes of triangle ABC from the vertices A, B, C, respectively. Denote by A 0 , B 0 , C 0 the midpoints of these altitudes AH, BK, CL, respectively. The incircle of triangle ABC has center I and touches the sides BC, CA, AB at the points D, E, F , respectively. Prove that the four lines A 0 D, B 0 E, C 0 F and OI are concurrent. (When the point O concides with I, we consider the line OI as an arbitrary line passing through O.) 3 Let f(0, 0) = 5 2003 , f(0, n) = 0 for every integer n = 0 and f(m, n) = f(m − 1, n) − 2 ·  f(m − 1, n) 2  +  f(m − 1, n − 1) 2  +  f(m − 1, n + 1) 2  for every natural number m > 0 and for every integer n. Prove that there exists natural number M such that f(M, n) = 1 for all integers n such that |n| ≤ (5 2003 − 1) 2 and f(M, n) = 0 for all integers n such that |n| > 5 2003 − 1 2 . (Here [x] denotes the integral part of real number x). http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ vnmath.com Vietnam Team Selection Tests 2003 Day 2 1 On the sides of triangle ABC take the points M 1 , N 1 , P 1 such that each line M M 1 , NN 1 , P P 1 divides the perimeter of ABC in two equal parts (M, N, P are respectively the midpoints of the sides BC, CA, AB). I. Prove that the lines M M 1 , NN 1 , P P 1 are concurrent at a point K. II. Prove that among the ratios KA BC , KB CA , KC AB there exist at least a ratio which is not less than 1 √ 3 . 2 Let A be the set of all permutations a = (a 1 , a 2 , . . . , a 2003 ) of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset S of the set {1, 2, . . . , 2003} such that {a k |k ∈ S} = S. For each a = (a 1 , a 2 , . . . , a 2003 ) ∈ A, let d(a) = 2003  k=1 (a k − k) 2 . I. Find the least value of d(a). Denote this least value by d 0 . II. Find all permutations a ∈ A such that d(a) = d 0 . 3 Let n be a positive integer. Prove that the number 2 n + 1 has no prime divisor of the form 8 · k − 1, where k is a positive integer. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ vnmath.com [...]... Γ2 again at Q Show that the midpoint of P Q lies on the line M C and the line P Q passes through a fixed point when M moves on Γ1 v n m a t {[Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=214 ] This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 http://www.mathlinks.ro/ Vietnam Team... ab999 99999cd , e.g 2005 × 9 Let {an } : an < C · n, {an } is increasing Prove that {an } contain infinite diamond 2005 v n m a t Compare with [url=http://www.mathlinks.ro/Forum/topic-15091.html]this problem.[/url] This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2006 Day 1 1 Given... segments a number not greater than m sothat every triangle whose three points are in the 2006 points given has the following property: Two of this triangle’s sides are put two equal numbers, and the other a greater number Find the minimum value of the ”good” number m This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 http://www.mathlinks.ro/... represented as sum of at most n elements (not necessarily different) from S Let a be greatest element from S Prove that there are positive integer k and integers b such that |Sn | = a · n + b for all n > k This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2005 Day 1 1 Let (I), (O)... choices of k chairs so that k students can sit on those and the condition is satisfied v n m a t h c 3 Find all functions f : Z → Z satisfying the condition: f (x3 + y 3 + z 3 ) = f (x)3 + f (y)3 + f (z)3 This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2005 Day 2 1 Let be given... m 3 The real sequence {an |n = 0, 1, 2, 3, } defined a0 = 1 and v Denote n an+1 = 1 2 An = an + 1 3 · an 3 3 · a2 − 1 n Prove that An is a perfect square and it has at least n distinct prime divisors This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2007 Day 1 1 Given two sets... (O; R) is a circle and M is a point then PM/(O) = OM 2 − R2 A 2 cos2 C 2 + cos2 B 2 cos2 cos2 A 2 C 2 + cos2 C 2 cos2 cos2 B 2 A 2 v n cos2 B 2 m cos2 a t 3 Given a triangle ABC Find the minimum of This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2007 Day 2 4 Find all continuous... be a regular 9−gon Let {A1 , A2 , , A9 } = S1 ∪ S2 ∪ S3 such that |S1 | = |S2 | = |S3 | = 3 Prove that there exists A, B ∈ S1 , C, D ∈ S2 , E, F ∈ S3 such that AB = CD = EF and A = B, C = D, E = F This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2008 Day 1 1 On the plane,... property: There exists a positive integer c which is not greater than such that 2 |s1 − s2 | = c for every pairs of arbitrary elements s1 , s2 ∈ S How many does a wanting set have at most are there ? This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2008 Day 2 1 Let m and n... a S1 = (x, y, z) ∈ M 3 | x, y, z have the same color and 2008| (x + y + z)} ; S2 = (x, y, z) ∈ M 3 | x, y, z hav v n m Prove that 2 |S1 | > |S2 | (where |X| denotes the number of elements in a set X) This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 http://www.mathlinks.ro/ Vietnam Team Selection Tests 2009 Day 1 1 Let an acute . {[Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .] http://www.artofproblemsolving.com/ This file was. [url=http://www.mathlinks.ro/Forum/topic-15091.html]this problem.[/url] http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks