Prove that for positive real numbers a,b,c the following inequality holds: a9bc+1+b9ca+1+c9ab+1≥a+b+c1+(a+b+c)2When does equality occur?
2021 Serbia JBMO TSTs
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Solve the following equation in natural numbers: x2=2y+2021z
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Two players play the following game: alternatively they write numbers 1 or 0 in the vertices of an n-gon. First player starts the game and wins if after any of his moves there exists a triangle, whose vertices are three consecutive vertices of the n-gon, such that the sum of numbers in it's vertices is divisible by 3. Second player wins if he prevents this. Determine which player has a winning strategy if: a) n=2019 b) n=2020 c) n=2021
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On sides AB and AC of an acute triangle ΔABC, with orthocenter H and circumcenter O, are given points P and Q respectively such that APHQ is a parallelogram. Prove the following equality: PB⋅PQQA⋅QO=2