MathDB

2

Finite group, N_{ABC} = N_{CBA}

G

be a finite group. For arbitrary sets

U, V, W \subset G

, denote by

N_{UVW}

the number of triples

(x, y, z) \in U \times V \times W

for which

xyz

is the unity . Suppose that

G

is partitioned into three sets

A, B

and

C

(i.e. sets

A, B, C

are pairwise disjoint and

G = A \cup B \cup C

). Prove that

N_{ABC}= N_{CBA}.

Determinant of a certain kind of matrix

Let

n > 1

be an odd positive integer and

A = (a_{ij})_{i, j = 1..n}

be the

n \times n

matrix with

a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.

Find

\det A

.

6

2

How many nonzero coefficients can a polynomial

P(x)

have if its coefficients are integers and

|P(z)| \le 2

for any complex number

z

of unit length?

Sequence of polynomials eventually has real roots

Let

f \ne 0

be a polynomial with real coefficients. Define the sequence

f_{0}, f_{1}, f_{2}, \ldots

of polynomials by

f_{0}= f

and

f_{n+1}= f_{n}+f_{n}'

for every

n \ge 0

. Prove that there exists a number

N

such that for every

n \ge N

, all roots of

f_{n}

are real.

5

2

Prove that the function is null

Let

n

be a positive integer and

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

be arbitrary integers. Suppose that a function

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

satisfies

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

whenever

k

and

l

are integers and

l \ne 0

. Prove that

f = 0

.

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

For each positive integer

k

, find the smallest number

n_{k}

for which there exist real

n_{k}\times n_{k}

matrices

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

, (2)

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

for all

1 \le i, j \le k

, and (3)

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

.

3

2

Find all good quadratic homogeneous polynomials

Call a polynomial

P(x_{1}, \ldots, x_{k})

good if there exist

2\times 2

real matrices

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

such that

P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).

Find all values of

k

for which all homogeneous polynomials with

k

variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

Exists c such that f(c) = c

Let

C

be a nonempty closed bounded subset of the real line and

f: C\to C

be a nondecreasing continuous function. Show that there exists a point

p\in C

such that f(p) \equal{} p. (A set is closed if its complement is a union of open intervals. A function

g

is nondecreasing if

g(x)\le g(y)

for all

x\le y

.)

2

Minimal and maximal possible rank of a matrix

Let

n\ge 2

be an integer. What is the minimal and maximal possible rank of an

n\times n

matrix whose

n^{2}

entries are precisely the numbers

1, 2, \ldots, n^{2}

?

Sum of fourth powers divisible by 29

Let

x

,

y

and

z

be integers such that

S = x^{4}+y^{4}+z^{4}

is divisible by 29. Show that

S

is divisible by

29^{4}

.

1

2

Quadratic polynomial, coefficients divisible by 5

Let

f

be a polynomial of degree 2 with integer coefficients. Suppose that

f(k)

is divisible by 5 for every integer

k

. Prove that all coefficients of

f

are divisible by 5.

f can be rotated/translated onto cf

Let

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

be a continuous function. Suppose that for any

c > 0

, the graph of

f

can be moved to the graph of

cf

using only a translation or a rotation. Does this imply that

f(x) = ax+b

for some real numbers

a

and

b

?

2007 IMC

Subcontests

Finite group, N_{ABC} = N_{CBA}

Determinant of a certain kind of matrix

How many nonzero coefficients can a polynomial

Sequence of polynomials eventually has real roots

Prove that the function is null

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

Find all good quadratic homogeneous polynomials

Exists c such that f(c) = c

Minimal and maximal possible rank of a matrix

Sum of fourth powers divisible by 29

Quadratic polynomial, coefficients divisible by 5

f can be rotated/translated onto cf