MathDB

1

Decomposing a polynomial into a sum of polynomials

1976(x+x^2+ \cdots +x^n)

is decomposed into a sum of polynomials of the form

a_1x + a_2x^2 + \cdots + a_nx^n

, where

a_1, a_2, \ldots , a_n

are distinct positive integers not greater than

n

. Find all values of

n

for which such a decomposition is possible.

8

1

Prove that there exist polynomials P and Q

Let

P

be a polynomial with real coefficients such that

P(x) > 0

if

x > 0

. Prove that there exist polynomials

Q

and

R

with nonnegative coefficients such that

P(x) = \frac{Q(x)}{R(x)}

if

x > 0.

7

1

Interval I=(0,1] - ISL 1976

Let

I = (0, 1]

be the unit interval of the real line. For a given number

a \in (0, 1)

we define a map

T : I \to I

by the formula if T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases}
Show that for every interval

J \subset I

there exists an integer

n > 0

such that

T^n(J) \cap J \neq \emptyset.

4

1

Prove the identity of sequence - ISL 1976

A sequence

(u_{n})

is defined by u_{0}=2 u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \textnormal{for } n=1,\ldots Prove that for any positive integer

n

we have

[u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}}

(where [x] denotes the smallest integer

\leq

x)

.

2

1

Prove that a_k ≤ k(n+1-k)/2 for every k

Let

a_0, a_1, \ldots, a_n, a_{n+1}

be a sequence of real numbers satisfying the following conditions:

a_0 = a_{n+1 }= 0,

|a_{k-1} - 2a_k + a_{k+1}| \leq 1 (k = 1, 2,\ldots , n). Prove that |a_k| \leq \frac{k(n+1-k)}{2} (k = 0, 1,\ldots ,n + 1).

1

Prove that AB = BC = CA - ISL 1976

Let

ABC

be a triangle with bisectors

AA_1,BB_1, CC_1

(

A_1 \in BC

, etc.) and

M

their common point. Consider the triangles

MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C

, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then

AB = BC = CA.

11

1

5^n

Prove that

5^n

has a block of

1976

consecutive

0's

in its decimal representation.

1976 IMO Shortlist

Subcontests

Decomposing a polynomial into a sum of polynomials

Prove that there exist polynomials P and Q

Interval I=(0,1] - ISL 1976

Prove the identity of sequence - ISL 1976

Prove that a_k ≤ k(n+1-k)/2 for every k

Prove that AB = BC = CA - ISL 1976

5^n