Let
![x_1,x_2,\ldots,x_n](/media/m/3/a/4/3a498672e3108eccaf28c3c29f3916e2.png)
be real numbers satisfying
![x_1^2+x_2^2+\ldots+x_n^2=1](/media/m/d/3/2/d32fa533455071e65fb1a995cf1afc7b.png)
. Prove that for every integer
![k\ge2](/media/m/a/1/3/a13f424c307e6dd7ce532c59a8a8b49d.png)
there are integers
![a_1,a_2,\ldots,a_n](/media/m/e/1/c/e1cadd08528b76b10be041b63c00aa8b.png)
, not all zero, such that
![|a_i|\le k-1](/media/m/b/f/5/bf5e78ba6191a2adcc29ff45729b19ac.png)
for all
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
, and
![|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}](/media/m/5/1/7/517aa57b0009a61e470ae5c02ed82682.png)
. (IMO Problem 3)
Proposed by Germany, FR
%V0
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