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42nd IMO 2001 shortlisted problems and solutions
Algebra A1. Let T be the set of all triples (a, b, c) where a, b, c are non-negative integers. Find all real-valued functions f on T such that f(a, b, c) = 0 if any of a, b, c are zero and f(a, b,... -
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41st IMO 2000 shortlisted problems and solutions
Algebra A2. a, b, c are positive integers such that b > 2a, c > 2b. Show that there is a real k such that the fractional parts of ka, kb, kc all exceed 1/3 and do not exceed 2/3. ... -
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40th IMO 1999 shortlisted problems and solutions
AlgebraA2. The numbers 1 to n2 are arranged in the squares of an n x n board (1 per square). There are n2 (n-1) pairs of numbers in the same row or column. For each such pair take the larger... -
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39th IMO 1998 shortlisted Problems and Solutions
Algebra A1. x1, x2, ... , xn are positive reals with sum less than 1. Show that nn+1x1x2 ... xn(1 - x1 - ... - xn) ≤ (x1 + x2 + ... + xn)(1 - x1)(1 - x2) ... (1 - xn). A2. x1, x2, ... ,... -
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38th IMO 1997 shortlisted Problems and Solutions
2. The sequences Rn are defined as follows. R1 = (1). If Rn = (a1, a2, ... , am), then Rn+1 = (1, 2, ... , a1, 1, 2, ... , a2, 1, 2, ... , 1, 2, ... , am, n+1). For example, R2 = (1, 2), R3 = (1, 1,... -
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37th IMO 1996 shortlisted Problems and Solutions
1. x, y, z are positive real numbers with product 1. Show that xy/(x5 + xy + y5) + yz/(y5 + yz + z5) + zx/(z5 + zx + x5) ≤ 1. When does equality occur? 2. x1 ≥ x2 ≥ ... ≥ xn are real... -
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34th IMO 1990 shortlisted Problems and Solutions
1. Show that there is a finite set of points in the plane such that for any point P in the set we can find 1993 points in the set a distance 1 from P. 2. ABC is a triangle with... -
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33rd IMO 1990 shortlisted Problems and Solutions
1. m is a positive integer. If there are two coprime integers a, b such that a divides m + b2 and b divides m + a2, show that we can also find two such coprime integers with the additional... -
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32nd IMO 1990 shortlisted Problems and Solutions
32nd IMO 1990 shortlisted Problems and Solutions 1. ABC is a triangle and P an interior point. Let the feet of the perpendiculars from P to AC, BC be P1, P2 respectively, and let the feet of the... -
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11th Chinese Mathematical Olympiad 1996 Problems & Solutions
A1. ABC is a triangle. The tangents from A touch the circle diameter BC at X and Y. Show that the orthocenter of ABC (point where the altitudes meet) lies on XY. A2. What is the... -
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10th Chinese Mathematical Olympiad 1995 Problems & Solutions
A1. Two sequences of positive reals are defined as follows. a1 = a2 = k. an+2 = an+1 + an. b2 ≥ b1, bn+2 >= bn+1 + bn. If a1 + a2 + ... + am = b1 + b2 + ... + bm for some m, show that am+1 ≤ bm... -
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9th Chinese Mathematical Olympiad 1994 Problems & Solutions
A1. ABCD is a convex quadrilateral. X lies on the segment AB and Y lies on the segment CD. W is the intersection of XC and BY, and Z is the intersection of AY and DX. If AB is parallel to CD ... -
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64th Putnam Mathematical Competition 2003 Problems
A1. Given n, how many ways can we write n as a sum of one or more positive integers a1 ≤ a2 ≤ ... ≤ ak with ak - a1 = 0 or 1. A2. a1, a2, ... , an, b1, ... , bn are non-negative... -
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63rd Putnam Mathematical Competition 2002 Problems
A1. k and n are positive integers. Let f(x) = 1/(xk - 1). Let p(x) = (xk - 1)n+1 fn(x), where fn is the nth derivative. Find p(1). A2. Given any 5 distinct points on the surface of... -
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62nd Putnam Mathematical Competition 2001 Problems
A1. Given a set X with a binary operation *, not necessarily associative or commutative, but such that (x * y) * x = y for all x, y in X. Show that x * (y * x) = y for all x, y in X. ... -
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45th Putnam Mathematical Competition 1984 Problems
A1. S is an a x b x c brick. T is the set of points a distance 1 or less from S. Find the volume of T. A2. Evaluate 6/( (9 - 4)(3 - 2) ) + 36/( (27 - 8)(9 - 4) ) + ... + 6n/( (3n+1... -
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44th Putnam Mathematical Competition 1983 Problems
A1. How many positive integers divide at least one of 1040 and 2030? A2. A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at... -
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43rd Putnam Mathematical Competition 1982 Problems
A1. Let S be the set of points (x, y) in the plane such that |x| ≤ y ≤ |x| + 3, and y ≤ 4. Find the position of the centroid of S. A2. Let Bn(x) = 1x + 2x + ... + nx and let f(n) =... -
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39th Putnam Mathematical Competition 1978 Problems
A1. Let S = {1, 4, 7, 10, 13, 16, ... , 100}. Let T be a subset of 20 elements of S. Show that we can find two distinct elements of T with sum 104. A2. Let A be the real n x n...
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ơ ơ, bài đấy tớ làm sai mà, bác nói thế tớ thấy ngại lắm :-">
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