China Western Mathematical Olympiad

Question

Let a_1, a_2, , a_n, b_1, b_2, , b_n be non-negative numbers satisfying the following conditions simultaneously: (1) _i=1^n (a_i + b_i) = 1; (2) _i=1^n i(a_i - b_i) = 0; (3) _i=1^n i^2 (a_i + b_i) = 10. Prove that a_k, b_k 1010+k^2 for all 1 k n.

Accepted Answer

For any 1 k n, it follows from the given conditions and Cauchy's Inequality that align* (ka_k)^2 & (_i=1^n i a_i)^2 = (_i=1^n i b_i)^2 & (_i=1^n i^2 b_i) (_i=1^n b_i) &= (10 - _i=1^n i^2 a_i) (1 - _i=1^n a_i) & (10 - k^2 a_k) (1 - a_k) &= 10 - (10 + k^2)a_k + k^2 a_k^2. align* It follows from that a_k 1010+k^2. Similarly, b_k 1010+k^2, and hence the result follows.