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Prove that there exists 3 integers satisfying the conditions

Source: India tst 2006 p11

June 27, 2012

modular arithmeticgeometry3D geometryanalytic geometrypigeonhole principlenumber theory unsolvednumber theory

Problem Statement

Let

u_{jk}

be a real number for each

j=1,2,3

and each

k=1,2

and let

N

be an integer such that

\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N

Let

M

and

l

be positive integers such that

l^2 <(M+1)^3

. Prove that there exist integers

\xi_1,\xi_2,\xi_3

not all zero, such that

\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}