5

Part of 2007 IMC

Problems(2)

Prove that the function is null

Source: IMC 2007, Day 1, Problem 5

n

be a positive integer and

a_{1}, \ldots, a_{n}

be arbitrary integers. Suppose that a function

f: \mathbb{Z}\to \mathbb{R}

\sum_{i=1}^{n}f(k+a_{i}l) = 0

k

l

are integers and

l \ne 0

f = 0

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How big do the matrices have to be to satisfy the properties

Source: IMC 2007 Day 2 Problem 5

For each positive integer

k

, find the smallest number

n_{k}

for which there exist real

n_{k}\times n_{k}

A_{1}, A_{2}, \ldots, A_{k}

such that all of the following conditions hold: (1)

A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0

A_{i}A_{j}= A_{j}A_{i}

1 \le i, j \le k

A_{1}A_{2}\ldots A_{k}\ne 0

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