2etang02
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7/26/2019 2etang02
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53-rd Mathematical Olympiad in Poland
Second Round, February 22–23, 2002
First Day
1. Prove that all functions f : R→ R satisfying
∀ x ∈ R f (x) = f (2x) = f (1 − x)
are periodic.
2. In a convex quadrangle ABCD the following equalities
<) ADB = 2<) ACB and <) BDC = 2<) BAC
hold. Prove that AD = C D.
3. A positive integer n is given. In an association consisting of n members work6 commissions. Each commission contains at least n/4 persons. Prove thatthere exist two commissions containing at least n/30 persons in common.
Second Day
4. Find all prime numbers p ≤ q ≤ r such that all the numbers
pq + r , pq + r2 , qr + p , qr + p2 , rp + q , rp + q 2
are prime.
5. Triangle ABC with <) BAC = 90◦ is the base of the pyramid ABCD. Moreoverit holds
AD = BD and AB = C D .
Prove that <) ACD ≥ 30◦.
6. Find all positive integers n such that for all real numbers x1, x2 . . . xn, y1, y2 . . . yn
the following inequalityx1x2 . . . xn + y1y2 . . . yn ≤
x21 + y2
1 ·
x22 + y2
1 · . . . ·
x2n
+ y2n
holds.
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