MathDB

1

Prove the fractional inequality - ISL 1971

\frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4,

where

a_i > 0, i = 1, 2, 3, 4.

15

1

Numbers from 11 to 99 are written on 99 cards

Natural numbers from

1

to

99

(not necessarily distinct) are written on

99

cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by

100

. Show that all the cards contain the same number.

14

1

Length of roken line is greater than 1248

A broken line

A_1A_2 \ldots A_n

is drawn in a

50 \times 50

square, so that the distance from any point of the square to the broken line is less than

1

. Prove that its total length is greater than

1248.

12

1

The midpoints either are collinear or form equilateral trgle

Two congruent equilateral triangles

ABC

and

A'B'C'

in the plane are given. Show that the midpoints of the segments

AA',BB', CC'

either are collinear or form an equilateral triangle.

11

1

Matrix satisfy the inequality, prove the other inequality

The matrix

A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}

satisfies the inequality

\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M

for each choice of numbers

x_i

equal to

\pm 1

. Show that

|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.

9

1

Show the identity for sequence T_n - ISL 1971

Let

T_k = k - 1

for

k = 1, 2, 3,4

and

T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).

Show that for all

k

, 1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \text{and} 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right], where

[x]

denotes the greatest integer not exceeding

x.

8

1

Determine whether there exist real numbers a, b, c,t - ISL71

Determine whether there exist distinct real numbers

a, b, c, t

for which:
(i) the equation

ax^2 + btx + c = 0

has two distinct real roots

x_1, x_2,

(ii) the equation

bx^2 + ctx + a = 0

has two distinct real roots

x_2, x_3,

(iii) the equation

cx^2 + atx + b = 0

has two distinct real roots

x_3, x_1.

6

1

Find a way to assign natural numbers - ISL 1971

Let

n \geq 2

be a natural number. Find a way to assign natural numbers to the vertices of a regular

2n

-gon such that the following conditions are satisfied:
(1) only digits

1

and

2

are used;
(2) each number consists of exactly

n

digits;
(3) different numbers are assigned to different vertices;
(4) the numbers assigned to two neighboring vertices differ at exactly one digit.

4

1

Find the length as a function of r_1 and r_2

We are given two mutually tangent circles in the plane, with radii

r_1, r_2

. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of

r_1

and

r_2

and the condition for the solvability of the problem.

3

1

Find the value of x^5+y^5+z^5

Knowing that the system

x + y + z = 3,

x^3 + y^3 + z^3 = 15,

x^4 + y^4 + z^4 = 35,

has a real solution

x, y, z

for which

x^2 + y^2 + z^2 < 10

, find the value of

x^5 + y^5 + z^5

for that solution.

1

Sequence of polynomials - ISL 1971

Consider a sequence of polynomials

P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots

, where

P_0(x) = 2, P_1(x) = x

and for every

n \geq 1

the following equality holds:

P_{n+1}(x) + P_{n-1}(x) = xP_n(x).

Prove that there exist three real numbers

a, b, c

such that for all

n \geq 1,

(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.

1971 IMO Shortlist

Subcontests

Prove the fractional inequality - ISL 1971

Numbers from 11 to 99 are written on 99 cards

Length of roken line is greater than 1248

The midpoints either are collinear or form equilateral trgle

Matrix satisfy the inequality, prove the other inequality

Show the identity for sequence T_n - ISL 1971

Determine whether there exist real numbers a, b, c,t - ISL71

Find a way to assign natural numbers - ISL 1971

Find the length as a function of r_1 and r_2

Find the value of x^5+y^5+z^5

Sequence of polynomials - ISL 1971