MathDB

Let

x_1,x_2,\ldots,x_n

be real numbers satisfying

x_1^2+x_2^2+\ldots+x_n^2=1

. Prove that for every integer

k\ge2

there are integers

a_1,a_2,\ldots,a_n

, not all zero, such that

|a_i|\le k-1

for all

i

, and

|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}

. (IMO Problem 3)
Proposed by Germany, FR

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