4

Part of 2001 IMC

Problems(2)

IMC 2001 Problem 4

Source: IMC 2001 Day 1 Problem 4

p(x)

is a polynomial of degree

n

with every coefficient

0

\pm1

p(x)

is divisible by

(x - 1)^k

for some integer

k > 0

q

is a prime such that

\frac{q}{\ln q}< \frac{k}{\ln n+1}

. Show that the complex

q

-th roots of unity must be roots of

p(x).

roots of unityPolynomials

IMC 2001 Problem 10

Source: IMC 2001 Day 2 Problem 4

A=(a_{k,l})_{k,l=1,...,n}

n \times n

matrix such that for each

m \in \{1,2,...,n\}

1 \leq j_{1} <...<j_{m}

the determinant of the matrix

(a_{j_{k},j_{l}})_{k,l=1,...,n}

is zero. Prove that

A^{n}=0

and that there exists a permutation

\sigma \in S_{n}

such that the matrix

(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}

has all of its nonzero elements above the diagonal.

linear algebramatrix