Home Competitions KMO 2013 Problem 7
For positive integer k k k { b n } , { c n } \{ b_n \}, \{ c_n \} { b n } , { c n } b 1 = c 1 = 1 b_1 = c_1 = 1 b 1 = c 1 = 1 b 2 n = k b 2 n − 1 + ( k − 1 ) c 2 n − 1 , c 2 n = b 2 n − 1 + c 2 n − 1 b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} b 2 n = k b 2 n − 1 + ( k − 1 ) c 2 n − 1 , c 2 n = b 2 n − 1 + c 2 n − 1 b 2 n + 1 = b 2 n + ( k − 1 ) c 2 n , c 2 n + 1 = b 2 n + k c 2 n b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} b 2 n + 1 = b 2 n + ( k − 1 ) c 2 n , c 2 n + 1 = b 2 n + k c 2 n a k = b 2014 a_k = b_{2014} a k = b 2014 ∑ k = 1 100 ( a k − a k 2 − 1 ) 1 2014 \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} } k = 1 ∑ 100 ( a k − a k 2 − 1 ) 2014 1
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