∑1≤i≤n (xi - yi)2 ≤ ∑1≤i≤n (xi - zi)2.
(1) P(tx, ty) = tnP(x, y) for some positive integer n and all real t, x, y;
(2) for all real x, y, z: P(y + z, x) + P(z + x, y) + P(x + y, z) = 0;
(3) P(1, 0) = 1.
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A1. Let x1 ≥ x2 ≥ ... ≥ xn, and y1 ≥ y2 ≥ ... ≥ yn be real numbers. Prove that if zi is any permutation of the yi, then:
∑1≤i≤n (xi - yi)2 ≤ ∑1≤i≤n (xi - zi)2. |
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| A2. Let a1 < a2 < a3 < ... be positive integers. Prove that for every i ≥ 1, there are infinitely many an that can be written in the form an = rai + saj, with r, s positive integers and j > i. |
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| A3. Given any triangle ABC, construct external triangles ABR, BCP, CAQ on the sides, so that ∠PBC = 45o, ∠PCB = 30o, ∠QAC = 45o, ∠QCA = 30o, ∠RAB = 15o, ∠RBA = 15o. Prove that ∠QRP = 90o and QR = RP. |
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| B1. Let A be the sum of the decimal digits of 44444444, and B be the sum of the decimal digits of A. Find the sum of the decimal digits of B. |
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| B2. Find 1975 points on the circumference of a unit circle such that the distance between each pair is rational, or prove it impossible. |
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B3. Find all polynomials P(x, y) in two variables such that:
(1) P(tx, ty) = tnP(x, y) for some positive integer n and all real t, x, y; (2) for all real x, y, z: P(y + z, x) + P(z + x, y) + P(x + y, z) = 0; (3) P(1, 0) = 1. |
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