1987 IMO Longlists

Part of IMO Longlists

Subcontests

(55)

1

IMO LongList 1987 - Find the least positive integer k

Find the least positive integer

k

such that for any

a \in [0, 1]

and any positive integer

n,

a^k(1 - a)^n < \frac{1}{(n+1)^3}.

1

IMO LongList 1987 - prove the existence of limit

Given two sequences of positive numbers

\{a_k\}

\{b_k\} \ (k \in \mathbb N)

a_k < b_k,

\cos a_kx + \cos b_kx \geq -\frac 1k

k \in \mathbb N

x \in \mathbb R,

prove the existence of

\lim_{k \to \infty} \frac{a_k}{b_k}

and find this limit.

1

IMO LongList 1987 - Inequality on 1987 positive reals

a_k

be positive numbers such that

a_1 \geq 1

a_{k+1} -a_k \geq 1 \ (k = 1, 2, . . . )

. Prove that for every

n \in \mathbb N,

\sum_{k=1}^{1987}\frac{1}{a_{k+1} \sqrt[1987]{a_k}} <1987

1

IMO LongList 1987 - Prove that there exist x, y

f(x)

be a periodic function of period

T > 0

\mathbb R

. Its first derivative is continuous on

\mathbb R

. Prove that there exist

x, y \in [0, T )

x \neq y

f(x)f'(y)=f'(x)f(y).

1

IMO LongList 1987 - Cover a rectangle with triminos

Is it possible to cover a rectangle of dimensions

m \times n

with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter

\text L

m \times n = 1985 \times 1987;

(b) m \times n = 1987 \times 1989 ?

1

IMO LongList 1987 - Sequence

To every natural number

k, k \geq 2

, there corresponds a sequence

a_n(k)

according to the following rule: a_0 = k, \qquad a_n = \tau(a_{n-1}) \forall n \geq 1, in which

\tau(a)

is the number of different divisors of

a

k

for which the sequence

a_n(k)

does not contain the square of an integer.

1

IMO LongList 1987 - Maximum of the expression

a, b, c, d

are real numbers such that

a^2 + b^2 + c^2 + d^2 \leq 1

, find the maximum of the expression

(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.

1

IMO LongList 1987 - Runs of a decimal number

The runs of a decimal number are its increasing or decreasing blocks of digits. Thus

024379

has three runs :

024, 43

379

. Determine the average number of runs for a decimal number in the set

\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}

n \geq 2.

1

IMO LongList 1987 - Base-r expansion

r > 1

be a real number, and let

n

be the largest integer smaller than

r

. Consider an arbitrary real number

x

0 \leq x \leq \frac{n}{r-1}.

r

x

we mean a representation of

x

x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots

a_i

are integers with

0 \leq a_i < r.

You may assume without proof that every number

x

0 \leq x \leq \frac{n}{r-1}

has at least one base-

r

expansion.
Prove that if

r

is not an integer, then there exists a number

p

0 \leq p \leq \frac{n}{r-1}

, which has infinitely many distinct base-

r

1

IMO LongList 1987 - Compute the sum

\sum_{k=0}^{2n} (-1)^k a_k^2

a_k

are the coefficients in the expansion

(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.

1

IMO LongList 1987 - Geometric inequality

l, l'

be two lines in

3

A,B,C

be three points taken on

l

B

as midpoint of the segment

AC

a, b, c

are the distances of

A,B,C

l'

, respectively, show that

b \leq \sqrt{ \frac{a^2+c^2}{2}}

, equality holding if

l, l'

1

IMO LongList 1987 - Sum of areas

PQ

be a line segment of constant length

\lambda

taken on the side

BC

ABC

B,P,Q,C

, and let the lines through

P

Q

parallel to the lateral sides meet

AC

P_1

Q_1

AB

P_2

Q_2

respectively. Prove that the sum of the areas of the trapezoids

PQQ_1P_1

PQQ_2P_2

is independent of the position of

PQ

BC.

1

IMO LongList 1987 - Find y (Proposed by Turkey)

It is given that

x = -2272

y = 10^3+10^2c+10b+a

z = 1

satisfy the equation

ax + by + cz = 1

a, b, c

are positive integers with

a < b < c

y.

1

IMO LongList 1987 - Complex and real numbers

It is given that

a_{11}, a_{22}

are real numbers, that

x_1, x_2, a_{12}, b_1, b_2

are complex numbers, and that

a_{11}a_{22}=a_{12}\overline{a_{12}}

\overline{a_{12}}

is he conjugate of

a_{12}

). We consider the following system in

x_1, x_2

\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,

\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.

(a) Give one condition to make the system consistent.
(b) Give one condition to make

\arg x_1 - \arg x_2 = 98^{\circ}.

1

IMO LongList 1987 SPA1 - (Ugly ?) equation to solve in Z

Find, with argument, the integer solutions of the equation

3z^2 = 2x^3 + 385x^2 + 256x - 58195.

1

IMO LongList 1987-B'C' intersects incircle in two points

The bisectors of the angles

B,C

ABC

intersect the opposite sides in

B', C'

respectively. Prove that the straight line

B'C'

intersects the inscribed circle in two different points.

1

The union of all straight lines - IMO LongList 1987

Two moving bodies

M_1,M_2

are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines

M_1M_2.

1

IMO LongList 1987 - Diophantine equation

n

be a natural number. Solve in integers the equation

x^n + y^n = (x - y)^{n+1}.

1

ILL 1987 - Mapping function

F

is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that

F

maps squares into squares.

1

IMO LongList 1987 Polynomials

P,Q,R

be polynomials with real coefficients, satisfying

P^4+Q^4 = R^2

. Prove that there exist real numbers

p, q, r

and a polynomial

S

P = pS, Q = qS

R = rS^2

.
[hide="Variants"]Variants. (1)

P^4 + Q^4 = R^4

\gcd(P,Q) = 1

\pm P^4 + Q^4 = R^2

R^4.

1

Convex polygon W

In the coordinate system in the plane we consider a convex polygon

W

and lines given by equations

x = k, y = m

k

m

are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of

W

intersects a square if the boundary contains an interior point of the square. Prove that the boundary of

W

intersects at most 4 \lceil d \rceil unit squares, where

d

is the maximal distance of points belonging to

W

(i.e., the diameter of

W

) and \lceil d \rceil is the least integer not less than

d.

1

Q,R,S are collinear

Through a point

P

within a triangle

ABC

l, m

n

perpendicular respectively to

AP,BP,CP

are drawn. Prove that if

l

intersects the line

BC

Q

m

AC

R

n

AB

S

, then the points

Q, R

S

1

Variable Polygon

Let us consider a variable polygon with

2n

n \in N

) in a fixed circle such that

2n - 1

of its sides pass through

2n - 1

fixed points lying on a straight line

\Delta

. Prove that the last side also passes through a fixed point lying on

\Delta .

1

Inequality with sin θ

\theta_1,\theta_2,\cdots,\theta_n

n

real numbers such that

\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0

|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]

1

Number of circles

2n + 3

points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly

n

interior points is not less than

\frac 13 \binom{2n+3}{2}.

1

Floor function solutions

Find the integer solutions of the equation

\left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right]

1

Minimum number of segments

n

points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?

1

Find the length of the side c

The perpendicular line issued from the center of the circumcircle to the bisector of angle

C

ABC

divides the segment of the bisector inside

ABC

into two segments with ratio of lengths

\lambda

b = AC

a = BC

, find the length of side

c.

1

Set of polynomials with alot of properties

A

be a set of polynomials with real coefficients and let them satisfy the following conditions:
(i) if

f \in A

\deg( f ) \leq 1

f(x) = x - 1

f \in A

\deg( f ) \geq 2

, then either there exists

g \in A

f(x) = x^{2+\deg(g)} + xg(x) -1

g, h \in A

f(x) = x^{1+\deg(g)}g(x) + h(x)

;
(iii) for every

g, h \in A

x^{2+\deg(g)} + xg(x) -1

x^{1+\deg(g)}g(x) + h(x)

A.

R_n(f)

be the remainder of the Euclidean division of the polynomial

f(x)

x^n

. Prove that for all

f \in A

and for all natural numbers

n \geq 1

R_n(f)(1) \leq 0

R_n(f)(1) = 0

R_n(f) \in A

1

Probablity is greater than 6/{(n-2)^3}

Five distinct numbers are drawn successively and at random from the set

\{1, \cdots , n\}

. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than

\frac{6}{(n-2)^3}

1

Determine the probablity

A game consists in pushing a flat stone along a sequence of squares

S_0, S_1, S_2, . . .

that are arranged in linear order. The stone is initially placed on square

S_0

. When the stone stops on a square

S_k

it is pushed again in the same direction and so on until it reaches

S_{1987}

or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly

n

\frac{1}{2^n}

. Determine the probability that the stone will stop exactly on square

S_{1987}.

1

Solve 28^x = 19^y +87^z in integers

Solve the equation

28^x = 19^y +87^z

x, y, z

1

Construct a triangle

Construct a triangle

ABC

a = BC

, its circumradius

R \ (2R \geq a)

, and the difference

\frac{1}{k} = \frac{1}{c}-\frac{1}{b}

c = AB

b = AC.

1

Prove that the sum is non zero

Consider the regular

1987

A_1A_2 . . . A_{1987}

O

. Show that the sum of vectors belonging to any proper subset of

M = \{OA_j | j = 1, 2, . . . , 1987\}

1

Chess players

In a chess tournament there are

n \geq 5

players, and they have already played

\left[ \frac{n^2}{4} \right] +2

games (each pair have played each other at most once).
(a) Prove that there are five players

a, b, c, d, e

for which the pairs

ab, ac, bc, ad, ae, de

have already played.
(b) Is the statement also valid for the

\left[ \frac{n^2}{4} \right] +1

games played?
Make the proof by induction over

n.

1

Find the smallest real number C

Find, with proof, the smallest real number

C

with the following property: For every infinite sequence

\{x_i\}

of positive real numbers such that

x_1 + x_2 +\cdots + x_n \leq x_{n+1}

n = 1, 2, 3, \cdots

\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.

1

Inequality with x^2+y^2+z^2=2

x, y, z

are real numbers such that

x^2+y^2+z^2 = 2

x + y + z \leq xyz + 2.

1

d(m,n) are integers

d(n,m)

m, n

0 \leq m \leq n

, are defined by

d(n, 0) = d(n, n) = 0

n \geq 0

md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.

Prove that all the

d(n,m)

1

Equation with degree 4

Prove that if the equation

x^4 + ax^3 + bx + c = 0

has all its roots real, then

ab \leq 0.

1

Lampshade

A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance

d

from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. Prove that the area of the lampshade is

d^2(2\theta + \sqrt 3)

\cot \frac {\theta}{2} = \frac{3}{\theta}.

1

Show that alpha is irrational

Consider the number

\alpha

obtained by writing one after another the decimal representations of

1, 1987, 1987^2, \dots

to the right the decimal point. Show that

\alpha

1

Find M such that B'C' is minimum

ABC

be a triangle. For every point

M

belonging to segment

BC

B'

C'

the orthogonal projections of

M

on the straight lines

AC

BC

M

for which the length of segment

B'C'

1

Interesting problem from ILL 1987 FRA2

a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3

be nine strictly positive real numbers. We set S_1 = a_1b_2c_3, S_2 = a_2b_3c_1, S_3 = a_3b_1c_2;T_1 = a_1b_3c_2, T_2 = a_2b_1c_3, T_3 = a_3b_2c_1. Suppose that the set

\{S1, S2, S3, T1, T2, T3\}

has at most two elements.
Prove that

S_1 + S_2 + S_3 = T_1 + T_2 + T_3.

1

n variables ILL 1987 inequality

n

0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1

(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.

1

Infinite set

A

be an infinite set of positive integers such that every

n \in A

is the product of at most

1987

prime numbers. Prove that there is an infinite set

B \subset A

p

such that the greatest common divisor of any two distinct numbers in

B

p.

1

Compute g(x,t) with the computer

S \subset [0, 1]

be a set of 5 points with

\{0, 1\} \subset S

. The graph of a real function

f : [0, 1] \to [0, 1]

is continuous and increasing, and it is linear on every subinterval

I

[0, 1]

such that the endpoints but no interior points of

I

S

.
We want to compute, using a computer, the extreme values of

g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}

x - t, x + t \in [0, 1]

. At how many points

(x, t)

is it necessary to compute

g(x, t)

with the computer?

1

There exist a constant c

In a Cartesian coordinate system, the circle

C_1

O_1(-2, 0)

3

. Denote the point

(1, 0)

A

and the origin by

O

.Prove that there is a constant

c > 0

such that for every

X

that is exterior to

C1

OX- 1 \geq c \min\{AX,AX^2\}.

Find the largest possible

c.

1

Probablity of CUBA JULY 1987

20

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z\}

we have made a random sequence of

28

throws. What is the probability that the sequence

CUBA \ JULY \ 1987

appears in this order in the sequence already thrown?

1

The least positive integer n

Determine the least possible value of the natural number

n

n!

ends in exactly

1987

zeros.
[hide="Note"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.

1

There exists a function u

f : (0,+\infty) \to \mathbb R

be a function having the property that

f(x) = f\left(\frac{1}{x}\right)

x > 0.

Prove that there exists a function

u : [1,+\infty) \to \mathbb R

u\left(\frac{x+\frac 1x }{2} \right) = f(x)

x > 0.

1

Collinear and area problem

Let there be given three circles

K_1,K_2,K_3

O_1,O_2,O_3

respectively, which meet at a common point

P

K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}

. Given an arbitrary point

X

K_1

X

A

K_2

Y

X

C

K_3

Z.

(a) Show that the points

Z,B, Y

are collinear. (b) Show that the area of triangle

XY Z

is less than or equal to

4

times the area of triangle

O_1O_2O_3.

1

Six variables inequality - ILL 1987

a_1, a_2, a_3, b_1, b_2, b_3

be positive real numbers. Prove that

(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)

and show that the two sides of the inequality are equal if and only if

\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.

1

Towns

A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop

X

Y \neq X

, then we shall say

Y

can be reached from

X

. We shall use the phrase

Y

X

when we wish to express that every bus stop from which the bus stop

X

can be reached is a bus stop from which the bus stop

Y

can be reached, and every bus stop that can be reached from

Y

can also be reached from

X

. A visitor to this town discovers that if

X

Y

are any two different bus stops, then the two sentences “

Y

can be reached from

X

Y

X

” have exactly the same meaning in this town. Let

A

B

be two bus stops. Show that of the following two statements, exactly one is true: (i)

B

can be reached from

A;

A

can be reached from

B.

1

A pack of 2n cards

Suppose we have a pack of

2n

cards, in the order

1, 2, . . . , 2n

. A perfect shuffle of these cards changes the order to

n+1, 1, n+2, 2, . . ., n- 1, 2n, n

; i.e., the cards originally in the first

n

positions have been moved to the places

2, 4, . . . , 2n

, while the remaining

n

cards, in their original order, fill the odd positions

1, 3, . . . , 2n - 1.

Suppose we start with the cards in the above order

1, 2, . . . , 2n

and then successively apply perfect shuffles. What conditions on the number

n

are necessary for the cards eventually to return to their original order? Justify your answer.
[hide="Remark"] Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.

1

Same remainder when divided by 3

x_1, x_2,\cdots, x_n

n

n = p + q

p

q

are positive integers. For

i = 1, 2, \cdots, n

S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}

(it is assumed that

x_{i+n }= x_i

i

m(a, b)

be the number of indices

i

S_i

leaves the remainder

a

T_i

leaves the remainder

b

3

a, b \in \{0, 1, 2\}

m(1, 2)

m(2, 1)

leave the same remainder when divided by

3.