1988 IMO Longlists

Part of IMO Longlists

Subcontests

(63)

1

Find some numbers whose product is a perfect square

n+1, n \geq 1

positive integers be formed by taking the product of

n

given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these

n+1

one can find some numbers whose product is a perfect square.

1

Polynomial and binomial coefficient

Given a natural number

n,

find all polynomials

P(x)

of degree less than

n

satisfying the following condition

\sum^n_{i=0} P(i) \cdot (-1)^i \cdot \binom{n}{i} = 0.

1

Is there a lower boundary for a fraction sum inequality?

p \geq 2

be a natural number. Prove that there exist an integer

n_0

\sum^{n_0}_{i=1} \frac{1}{i \cdot \sqrt[p]{i + 1}} > p.

1

A regular 14-gon with side a is inscribed in a circle

A regular 14-gon with side

a

is inscribed in a circle of radius one. Prove

\frac{2-a}{2 \cdot a} > \sqrt{3 \cdot \cos \left( \frac{\pi}{7} \right)}.

1

Can we find this particular infinite sequence?

Does there exist a number

\alpha, 0 < \alpha < 1

such that there is an infinite sequence

\{a_n\}

of positive numbers satisfying

1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots?

1

We match sets M of points in the coordinate plane

M

of points in the coordinate plane to sets

M*

according to the rule that

(x*,y*) \in M*

if and only if x \cdot x* \plus{} y \cdot y* \leq 1 whenever

(x,y) \in M.

Find all triangles

Q

Q*

is the reflection of

Q

1

Each tangent to the fixed circle

Seven circles are given. That is, there are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two other adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order,

A_1, A_2, A_3, A_4, A_5

A_6

A_1 A_2 \cdot A_3 A_4 \cdot A_5 A_6 = A_2 A_3 \cdot A_4 A_5 \cdot A_6 A_1.

1

Satisfy equation bc - ad = 1

In a row written in increasing order all the irreducible positive rational numbers, such that the product of the numerator and the denominator is less than 1988. Prove that any two adjacent fractions

\frac{a}{b}

\frac{c}{d},

\frac{a}{b} < \frac{c}{d},

satisfy the equation

b \cdot c - a \cdot d = 1.

1

ax^2 + by^2 + cz^2 = 1

a,b,c

be integers different from zero. It is known that the equation

a \cdot x^2 + b \cdot y^2 + c \cdot z^2 = 0

(x,y,z)

in integer numbers different from the solutions

x = y = z = 0.

Prove that the equation

a \cdot x^2 + b \cdot y^2 + c \cdot z^2 = 1

has a solution in rational numbers.

1

Congruent inscribed circles

ABC

has a right angle at

C.

P

is located on segment

AC

such that triangles

PBA

PBC

have congruent inscribed circles. Express the length

x = PC

a = BC, b = CA

c = AB.

1

Job openings at a factory

n \geq 3

job openings at a factory, ranked

1

n

in order of increasing pay. There are

n

job applicants, ranked from

1

n

in order of increasing ability. Applicant

i

is qualified for job

j

i \geq j.

The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified AND which is lower in rank than any job already filled. (Under these rules, job

1

is always filled, and hiring terminates thereafter.) Show that applicants

n

and n \minus{} 1 have the same probability of being hired.

1

Partitioning can be carried out in unique manner

It is proposed to partition a set of positive integers into two disjoint subsets

A

B

subject to the conditions i.) 1 is in

A

ii.) no two distinct members of

A

have a sum of the form 2^k \plus{} 2, k \equal{} 0,1,2, \ldots; and iii.) no two distinct members of B have a sum of that form. Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.

1

We do not obtain any new sets

S

be an infinite set of integers containing zero, and such that the distances between successive number never exceed a given fixed number. Consider the following procedure: Given a set

X

of integers we construct a new set consisting of all numbers

x \pm s,

x

X

and s belongs to

S.

S_0 = \{0\}

we successively construct sets

S_1, S_2, S_3, \ldots

using this procedure. Show that after a finite number of steps we do not obtain any new sets, i.e.

S_k = S_{k_0}

k \geq k_0.

1

Centrebox and the cornerboxes contain one black stone

A two-person game is played with nine boxes arranged in a

3 \times 3

square and with white and black stones. At each move a player puts three stones, not necessarily of the same colour, in three boxes in either a horizontal or a vertical line. No box can contain stones of different colours: if, for instance, a player puts a white stone in a box containing black stones the white stone and one of the black stones are removed from the box. The game is over when the centrebox and the cornerboxes contain one black stone and the other boxes are empty. At one stage of a game

x

boxes contained one black stone each and the other boxes were empty. Determine all possible values for

x.

1

Collection are brought into coincidence

h+1

chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of

h?

1

Cyclic quadrilateral with cosecant equality

The quadrilateral

A_1A_2A_3A_4

is cyclic, and its sides are

a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4

a_4 = A_4A_1.

The respective circles with centres

I_i

r_i

are tangent externally to each side

a_i

and to the sides

a_{i+1}

a_{i-1}

a_0 = a_4

\prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2.

1

Spanish Sines Inradius Circumradius Inequality

ABC

is a triangle, with inradius

r

and circumradius

R.

\sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}.

1

They are seated around a table

n

people, each one knows exactly three others. They are seated around a table. We say that the seating is

perfect

if everyone knows the two sitting by their sides. Show that, if there is a perfect seating

S

for the group, then there is always another perfect seating which cannot be obtained from

S

by rotation or reflection.

1

Two primes p less than 20,000

For each positive integer

k

n,

S_k(n)

k

n.

Prove that there are at most two primes

p

20,000

S_{31}(p)

are composite numbers with at least two distinct prime divisors.

1

Construct a solid with 14 faces by cutting off all 8 corners

C

be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners at

C,

keeping the new faces perpendicular to the diagonals of the cube, and keeping the newly formed faces indentical. If at the conclusion of this process the fourteen faces so have the same area, find the area of each of face of the new solid.

1

GCD of the 1960-th and 1988-th terms of Fibonacci sequence

The Fibonacci sequence is defined by

a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1.

Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.

1

x^2 - 10*x - 22

Find all positive integers

x

such that the product of all digits of

x

x^2 - 10 \cdot x - 22.

1

Unique solution of the system of linear equations

x = p, y = q, z = r, w = s

be the unique solution of the system of linear equations

x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4.

Express the solutions of the following system in terms of

p,q,r

s:

x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4.

Assume the uniquness of the solution.

1

Summed up product divisible by 1001

a_1, \ldots, a_{10},

prove that there exist a non-zero sequence

\{x_1, \ldots, x_{10}\}

x_i

\{-1,0,1\}

\sum^{10}_{i=1} x_i \cdot a_i

is divisible by 1001.

1

Distance from it to O does not exceed 574

3

-dimensional space there is given a point

O

and a finite set

A

of segments with the sum of lengths equal to

1988

. Prove that there exists a plane disjoint from

A

such that the distance from it to

O

does not exceed

574

1

Sharp bounds for the ratio

For a convex polygon

P

in the plane let

P'

denote the convex polygon with vertices at the midpoints of the sides of

P.

Given an integer

n \geq 3,

determine sharp bounds for the ratio

\frac{\text{area } P'}{\text{area } P},

over all convex

n

P.

1

Any two elements have distance apart greater than or equal 3

S

is the set of all sequences \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}. The distance between two elements

\{a_i\}

\{b_i\}

S

is defined as \sum^7_{i \equal{} 1} |a_i \minus{} b_i|.

T

S

in which any two elements have a distance apart greater than or equal to 3. Prove that

T

contains at most 16 elements. Give an example of such a subset with 16 elements.

1

Points of a subset with 1788 points are coloured blue

Given a set of 1988 points in the plane. No four points of the set are collinear. The points of a subset with 1788 points are coloured blue, the remaining 200 are coloured red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100 red points.

1

IMO LongList 1988 Trignometric Inequality

\alpha_i > 0, \beta_i > 0

1 \leq i \leq n, n > 1

\sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi.

\sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i).

1

Conditions for sides to be inscribed in a circle

n

A_1, A_2, \ldots, A_n,

no three collinear, show that the

n

A_1 A_2 \ldots A_n,

is inscribed in a circle if and only if

A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots

+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n

= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n

XY

denotes the length of the segment

XY.

1

ABCD is a cyclic quadrilateral

ABCD

is a quadrilateral.

A'BCD'

is the reflection of

ABCD

BC,

A''B'CD'

is the reflection of

A'BCD'

CD'

A''B''C'D'

is the reflection of

A''B'CD'

D'A''.

Show that; if the lines

AA''

BB''

are parallel, then ABCD is a cyclic quadrilateral.

1

Twenty-nine of them form an arithmetic progression

The positive integer

n

has the property that, in any set of

n

integers, chosen from the integers

1,2, \ldots, 1988,

twenty-nine of them form an arithmetic progression. Prove that

n > 1788.

1

Rectangle Covering

Prove that the numbers

A,B

C

are equal, where: -

A=

number of ways that we can cover a

2 \times n

2 \times 1

B=

number of sequences of ones and twos that add up to

n

C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}

n = 2 \cdot m,

C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}

n = 2 \cdot m + 1.

1

29 different sequences of positive integers

A_1, A_2, \ldots, A_{29}

29

different sequences of positive integers. For

1 \leq i < j \leq 29

and any natural number

x,

N_i(x) =

number of elements of the sequence

A_i

which are less or equal to

x,

N_{ij}(x) =

number of elements of the intersection

A_i \cap A_j

which are less than or equal to

x.

It is given for all

1 \leq i \leq 29

and every natural number

x,

N_i(x) \geq \frac{x}{e},

e = 2.71828 \ldots

Prove that there exist at least one pair

i,j

1 \leq i < j \leq 29

N_{ij}(1988) > 200.

1

Deduce cosecant exression

-1 < x < 1.

\sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}.

\csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8.

1

g(n+2) = g(n) + g(n+1) + 1

g(n)

be defined as follows:

g(1) = 0, g(2) = 1

g(n+2) = g(n) + g(n+1) + 1, n \geq 1.

n > 5

is a prime, then

n

g(n) \cdot (g(n) + 1).

1

Plane triangles whose sides have integer length

Find all plane triangles whose sides have integer length and whose incircles have unit radius.

1

Indian Circle Problem

i.) Consider a circle

K

AB;

L

AB

K

and with a circle

M

tangent to circle

K,

L

AB.

Calculate the ration of the area of circle

K

to the area of circle

M.

ii.) In triangle

ABC, AB = AC

\angle CAB = 80^{\circ}.

D,E

F

BC, AC

AB,

respectively and

CE = CD

BF = BD,

then find the size of

\angle EDF.

1

Master of fractions for the new century wanted

x

x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})}

ii.) For each positive number

x,

k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)}

Calculate the minimum value of

k.

1

remainder r and numbers 1059, 1417 and 2312

i.) The polynomial

x^{2 \cdot k} + 1 + (x+1)^{2 \cdot k}

is not divisible by

x^2 + x + 1.

Find the value of

k.

p,q

r

are distinct roots of

x^3 - x^2 + x - 2 = 0

the find the value of

p^3 + q^3 + r^3.

r

is the remainder when each of the numbers 1059, 1417 and 2312 is divided by

d,

d

is an integer greater than one, then find the value of

d-r.

iv.) What is the smallest positive odd integer

n

such that the product of

2^{\frac{1}{7}}, 2^{\frac{3}{7}}, \ldots, 2^{\frac{2 \cdot n + 1}{7}}

is greater than 1000?

1

Polynomial g(x) = x^5 + x^4 + x^3 + x^2 + x + 1

g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.

What is the remainder when the polynomial

g(x^{12}

is divided by the polynomial

g(x)

k

is a positive number and

f

is a function such that, for every positive number

x, f(x^2 + 1 )^{\sqrt{x}} = k.

Find the value of

f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }}

for every positive number

y.

iii.) The function

f

satisfies the functional equation

f(x) + f(y) = f(x+y) - x \cdot y - 1

x,y

of real numbers. If

f(1) = 1,

then find the numbers of integers

n,

f(n) = n.

1

Four balls of radius 1 are mutually tangent

i.) Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length

s,

is circumscribed around the balls. Find the value of

s.

ii.) Suppose that

ABCD

EFGH

are opposite faces of a retangular solid, with \angle DHC \equal{} 45^{\circ} and \angle FHB \equal{} 60^{\circ}. Find the cosine of

\angle BHD.

1

Indian IMO LongList 1988 Geometry Problem

ABC

be a triangle with

AB = 12

AC = 16.

M

is the midpoint of side

BC

E

F

are chosen on sides

AC

AB

, respectively, and suppose that lines

EF

AM

G.

AE = 2 \cdot AF

then find the ratio

\frac{EG}{GF}

E

be a point external to a circle and suppose that two chords

EAB

EDC

meet at angle of

40^{\circ}.

AB = BC = CD

find the size of angle

ACD.

1

Sequence from Iceland

A sequence of numbers

a_n, n = 1,2, \ldots,

is defined as follows:

a_1 = \frac{1}{2}

n \geq 2

a_n = \frac{2 n - 3}{2 n} a_{n-1}.

\sum^n_{k=1} a_k < 1

n \geq 1.

1

Hungarian multiple choice test

In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any three of them there was a question which the three students answered differently. What is the maximum number of students tested?

1

n points are given on the surface of a sphere

n

points are given on the surface of a sphere. Show that the surface can be divided into

n

congruent regions such that each of them contains exactly one of the given points.

1

Mid-points of the three sides X,Y and Z

In the triangle

ABC

D,E

F

be the mid-points of the three sides,

X,Y

Z

the feet of the three altitudes,

H

the orthocenter, and

P,Q

R

the mid-points of the line segment joining

H

to the three vertices. Show that the nine points

D,E,F,P,Q,R,X,Y,Z

lie on a circle.

1

1988 as the sum of some positive integers

Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.

1

x^n + (2+x)^n + (2-x)^n = 0

Find a necessary and sufficient condition on the natural number

n

for the equation x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0 to have a integral root.

1

Condition for the roots to be exactly the cosines

Assuming that the roots of

x^3 + p \cdot x^2 + q \cdot x + r = 0

are real and positive, find a relation between

p,q

r

which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.

1

Coordinate axis and prime numbers

x^2+ y^2 = r^2

meets the coordinate axis at

A = (r,0), B = (-r,0), C = (0,r)

D = (0,-r).

P = (u,v)

Q = (-u,v)

be two points on the circumference of the circle. Let

N

be the point of intersection of

PQ

y

M

be the foot of the perpendicular drawn from

P

x

r^2

u = p^m > q^n = v,

p

q

are prime numbers and

m

n

are natural numbers, show that

|AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8.

1

Find the total number of different integers of a function

Find the total number of different integers the function

f(x) = \left[x \right] + \left[2 \cdot x \right] + \left[\frac{5 \cdot x}{3} \right] + \left[3 \cdot x \right] + \left[4 \cdot x \right]

0 \leq x \leq 100.

1

Sum equals Product

Find the positive integers

x_1, x_2, \ldots, x_{29}

at least one of which is greater that 1988 so that

x^2_1 + x^2_2 + \ldots x^2_{29} = 29 \cdot x_1 \cdot x_2 \ldots x_{29}.

1

Two perpendicular chords of a circle with centre O

CD

be two perpendicular chords of a circle with centre

O

r

X,Y,Z,W

denote the cyclical order of the four parts into which the disc is thus divided. Find the maximum and minimum of the quantity

\frac{A(X) + A(Z)}{A(Y) + A(W)},

A(U)

denotes the area of

U.

1

East German subsets

Z_{m,n}

be the set of all ordered pairs

(i,j)

i \in {1, \ldots, m}

j \in {1, \ldots, n}.

a_{m,n}

be the number of all those subsets of

Z_{m,n}

that contain no 2 ordered pairs

(i_1,j_1)

(i_2,j_2)

|i_1 - i_2| + |j_1 - j_2| = 1.

Then show, for all positive integers

m

k,

a^2_{m, 2 \cdot k} \leq a_{m, 2 \cdot k - 1} \cdot a_{m, 2 \cdot k + 1}.

1

Two similar problems proposed to IMO by Germany

n

runs through all the positive integers, then f(n) \equal{} \left \lfloor n \plus{} \sqrt {3n} \plus{} \frac {1}{2} \right \rfloor runs through all positive integers skipping the terms of the sequence a_n \equal{} \left \lfloor \frac {n^2 \plus{} 2n}{3} \right \rfloor.

1

Skipping the terms of a sequence

n

runs through all the positive integers, then f(n) \equal{} \left[n \plus{} \sqrt {\frac {n}{3}} \plus{} \frac {1}{2} \right] runs through all positive integers skipping the terms of the sequence a_n \equal{} 3 \cdot n^2 \minus{} 2 \cdot n.

1

All finite sequences of positive integers

1 \leq k \leq n.

Consider all finite sequences of positive integers with sum

n.

T(n,k),

the total number of terms of size

k

in all of the sequences.

1

Circumscribed about the same circle C

T

be a triangle with inscribed circle

C.

A square with sides of length

a

is circumscribed about the same circle

C.

Show that the total length of the parts of the edge of the square interior to the triangle

T

2 \cdot a.

1

27 half-lines (or rays) emanating from the origin

Show that there do not exist more than

27

half-lines (or rays) emanating from the origin in the

3

-dimensional space, such that the angle between each pair of rays is

\geq \frac{\pi}{4}

1

IMO LongList 1988 sequence

a_0

is a positive real number, consider the sequence

\{a_n\}

a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0.

Show that there exist a real number

a > 0

such that: i.) for all

a_0 \geq a,

\{a_n\} \rightarrow \infty,

a_0 < a,

\{a_n\} \rightarrow 0.

1

Partition M_k into two subsets

k

be a positive integer and

M_k

the set of all the integers that are between

2 \cdot k^2 + k

2 \cdot k^2 + 3 \cdot k,

both included. Is it possible to partition

M_k

A

B

\sum_{x \in A} x^2 = \sum_{x \in B} x^2.

1

there are infinitely many positive integers m

\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots,

[x]

denotes the integer part of

x.

Prove that i.) there are infinitely many positive integers

m

a_{m+1} - a_m > 1;

ii.) there are infinitely many positive integers

m

a_{m+1} - a_m = 1.