罗马尼亚IMO国家队选拔考试2000
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2. Prove that if f, g are monic polynomials with complex coefficients and f (f (x)) = g(g(x)), then f (x) = g(x). Marius Cavachi a3 + b3
3. Prove that every positive rational number can be represented in the form c3 + d3 , where a, b, c, d are positive integers. ISL 1999
Radu Todor
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
2nd Test - May 14, 2000 1. Prove that there exist infinitely many systems of positive integers (x, y, z, y) which do
|f (k + 1) − f (k)| ≥ 3, for all k ∈ {1, 2, . . . , n − 1}.
Vasile Pop
2. Let n ≥ 1 be a positive integer and x1, x2, . . . , xn be real numbers such that |xk+1 − xk| ≤ 1, for all k ∈ {1, 2, . . . , n − 1}. Prove that
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
Romanian Olympiad 2000 IMO Team Selection Tests
1st Test - April 27, 2000
1. Let n ≥ 2 be a positive integer. Find the number of functions f : {1, 2, . . . , n} → {1, 2, 3, 4, 5} which have the following property
n = ± a1 ± a2 ± a3 ± a4 ± a5 .
3
3
3
3
3
Radu Ignat
4. Let P1P2 . . . Pn be a convex polygon in the plane. We assume that for any arbitrary choice of vertices Pi, Pj there exists a vertex in the polygon Pk, such that ∠PiPkPj = 60◦. Prove that n = 3.
2. Let ABC be an acute triangle and M be the midpoint of the line segment BC. Consider the interior point N such that ∠ABN = ∠BAM and ∠CAN = ∠CAM . Prove that ∠BAN = ∠CAM . Gabriel Nagy
3. Let S be the set of interior points of a sphere and C be the set of interior points of a circle. Find, with proof, if there exists or not a function f : S → C such that |AB| ≤ |f (A)f (B)|, for any points A, B ∈ S, where |XY | denotes here the Euclidian distance between the points X and Y . Marius Cavachi
n
n
n2 − 1
|xk| −
xk ≤
. 4
k=1
k=1
Gh. Eckstein
3. Let n, k be arbitrary positive integers. Show that there exist positive integers
a1 > a2 > a3 > a4 > a5 > k such that
3. Determine all pairs m, n of positive integers such that any m × n rectangle can be tiled with congruent pieces consisting of three unit squares arranged in a L-like shape, like below
not have a common divisor greater than 1 and such that x3 + y3 + z2 = t4. Mihai Piticari and Sorin R˘adulescu
2. Let ABC be a triangle and M be an interior point. Prove that min{M A, M B, M C} + M A + M B + M C < AB + BC + CA. ISL 1999
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
4th Test - May 21, 2000
1. Let n ≥ 3 be an odd integer and m ≥ n2 − n + 1 be an integer number. The sequence of polygons P1, P2, . . . , Pm is defined as follows (i) P1 is a regular polygon with n vertices; (ii) Pk+1 is a regular polygon whose vertices are the midpoints of the sides of Pk, for any k ∈ {1, 2, . . . , m − 1}. Find, with proof, the maximum number of colors which can be used to color all the vertices of these polygons such that, for each such coloring, one could find four vertices A, B, C, D which have the same color and the quadrilateral ABCD is an isosceles trapezoid. Radu Ignat
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
3rd Test - May 20, Biblioteka Baidu000
1. Let a > 1 be an odd positive integer. Find the least positive integer n such that 22000 is a divisor of an − 1. Mircea Becheanu
3. Prove that every positive rational number can be represented in the form c3 + d3 , where a, b, c, d are positive integers. ISL 1999
Radu Todor
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
2nd Test - May 14, 2000 1. Prove that there exist infinitely many systems of positive integers (x, y, z, y) which do
|f (k + 1) − f (k)| ≥ 3, for all k ∈ {1, 2, . . . , n − 1}.
Vasile Pop
2. Let n ≥ 1 be a positive integer and x1, x2, . . . , xn be real numbers such that |xk+1 − xk| ≤ 1, for all k ∈ {1, 2, . . . , n − 1}. Prove that
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
Romanian Olympiad 2000 IMO Team Selection Tests
1st Test - April 27, 2000
1. Let n ≥ 2 be a positive integer. Find the number of functions f : {1, 2, . . . , n} → {1, 2, 3, 4, 5} which have the following property
n = ± a1 ± a2 ± a3 ± a4 ± a5 .
3
3
3
3
3
Radu Ignat
4. Let P1P2 . . . Pn be a convex polygon in the plane. We assume that for any arbitrary choice of vertices Pi, Pj there exists a vertex in the polygon Pk, such that ∠PiPkPj = 60◦. Prove that n = 3.
2. Let ABC be an acute triangle and M be the midpoint of the line segment BC. Consider the interior point N such that ∠ABN = ∠BAM and ∠CAN = ∠CAM . Prove that ∠BAN = ∠CAM . Gabriel Nagy
3. Let S be the set of interior points of a sphere and C be the set of interior points of a circle. Find, with proof, if there exists or not a function f : S → C such that |AB| ≤ |f (A)f (B)|, for any points A, B ∈ S, where |XY | denotes here the Euclidian distance between the points X and Y . Marius Cavachi
n
n
n2 − 1
|xk| −
xk ≤
. 4
k=1
k=1
Gh. Eckstein
3. Let n, k be arbitrary positive integers. Show that there exist positive integers
a1 > a2 > a3 > a4 > a5 > k such that
3. Determine all pairs m, n of positive integers such that any m × n rectangle can be tiled with congruent pieces consisting of three unit squares arranged in a L-like shape, like below
not have a common divisor greater than 1 and such that x3 + y3 + z2 = t4. Mihai Piticari and Sorin R˘adulescu
2. Let ABC be a triangle and M be an interior point. Prove that min{M A, M B, M C} + M A + M B + M C < AB + BC + CA. ISL 1999
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
4th Test - May 21, 2000
1. Let n ≥ 3 be an odd integer and m ≥ n2 − n + 1 be an integer number. The sequence of polygons P1, P2, . . . , Pm is defined as follows (i) P1 is a regular polygon with n vertices; (ii) Pk+1 is a regular polygon whose vertices are the midpoints of the sides of Pk, for any k ∈ {1, 2, . . . , m − 1}. Find, with proof, the maximum number of colors which can be used to color all the vertices of these polygons such that, for each such coloring, one could find four vertices A, B, C, D which have the same color and the quadrilateral ABCD is an isosceles trapezoid. Radu Ignat
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
3rd Test - May 20, Biblioteka Baidu000
1. Let a > 1 be an odd positive integer. Find the least positive integer n such that 22000 is a divisor of an − 1. Mircea Becheanu