Home Competitions ELMO SL 2012 A9
Let a , b , c a,b,c a , b , c k k k 3 3 3 ∣ a k + 1 ( b − c ) + b k + 1 ( c − a ) + c k + 1 ( a − b ) a k ( b − c ) + b k ( c − a ) + c k ( a − b ) ∣ ≥ k + 1 3 ( k − 1 ) ( a + b + c ) \left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c) a k ( b − c ) + b k ( c − a ) + c k ( a − b ) a k + 1 ( b − c ) + b k + 1 ( c − a ) + c k + 1 ( a − b ) ≥ 3 ( k − 1 ) k + 1 ( a + b + c ) ∣ a k + 2 ( b − c ) + b k + 2 ( c − a ) + c k + 2 ( a − b ) a k ( b − c ) + b k ( c − a ) + c k ( a − b ) ∣ ≥ ( k + 1 ) ( k + 2 ) 3 k ( k − 1 ) ( a 2 + b 2 + c 2 ) . \left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2). a k ( b − c ) + b k ( c − a ) + c k ( a − b ) a k + 2 ( b − c ) + b k + 2 ( c − a ) + c k + 2 ( a − b ) ≥ 3 k ( k − 1 ) ( k + 1 ) ( k + 2 ) ( a 2 + b 2 + c 2 ) .
Calvin Deng.
Dataset text AI-MO AoPS . Quoted with source.
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