1977 IMO Longlists

Part of IMO Longlists

Subcontests

(44)

1

Show that at least one of the numbers is greater than n!/2^n

x_0, x_1, \ldots , x_n

are integers and

x_0 > x_1 > \cdots > x_n.

Prove that at least one of the numbers

|F(x_0)|, |F(x_1)|, |F(x_2)|, \ldots, |F(x_n)|,

where F(x) = x^n + a_1x^{n-1} + \cdots+ a_n, a_i \in \mathbb R, i = 1, \ldots , n, is greater than

\frac{n!}{2^n}.

1

Nice geometric inequality of IMO LongList 1977

Prove that for every triangle the following inequality holds:

\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.

a, b, c

are lengths of the sides and

S

is the area of the triangle.

1

Faces of the tetrahedron are congruent triangles-[ILL 1977]

The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.

1

Prove the inequality for a ≤ b ≤ c ≤ d - [ILL 1977]

0 \leq a \leq b \leq c \leq d,

a^bb^cc^dd^a \geq b^ac^bd^ca^d.

1

Diogonals AO, BP and DN are concurrent

Through a point

O

on the diagonal

BD

of a parallelogram

ABCD

MN

AB

PQ

AD

, are drawn, with

M

AD

Q

AB

. Prove that diagonals

AO,BP,DN

(extended if necessary) will be concurrent.

1

Unlikely diophantine [ILL 1977]

Find all pairs of integers

a

b

7a+14b=5a^2+5ab+5b^2

1

Maximum and the minimum of the sum of the lengths [ILL 1977]

Two perpendicular chords are drawn through a given interior point

P

of a circle with radius

R.

Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from

P

to the center of the circle is

kR.

1

1.49 long segment [ILL 1977]

Several segments, which we shall call white, are given, and the sum of their lengths is

1

. Several other segments, which we shall call black, are given, and the sum of their lengths is

1

. Prove that every such system of segments can be distributed on the segment that is

1.51

long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is

1.49

1

Find all positive integers n - [ILL 1977]

Determine all positive integers

n

for which there exists a polynomial

P_n(x)

n

with integer coefficients that is equal to

n

n

different integer points and that equals zero at zero.

1

Swithcing coefficients give integer roots [ILL 1977]

Find all pairs of integers

(p,q)

for which all roots of the trinomials

x^2+px+q

x^2+qx+p

1

Area of AMNQ is maximal [ILL 1977]

ABCD

is given. A line passing through

A

CD

Q

. Draw a line parallel to

AQ

that intersects the boundary of the square at points

M

N

such that the area of the quadrilateral

AMNQ

1

Prove that 2a ≤ P ≤ 3a - [ILL 1977]

The intersection of a plane with a regular tetrahedron with edge

a

is a quadrilateral with perimeter

P.

2a \leq P \leq 3a.

1

Evalute the sum S [ILL 1977]

S=\sum_{k=1}^n k(k+1)\ldots (k+p),

n

p

are positive integers.

1

Interesting inequality on functions - [ILL 1977]

f

be a strictly increasing function defined on the set of real numbers. For

x

t

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

Assume that the inequalities

2^{-1} < g(x, t) < 2

hold for all positive t if

x = 0

t \leq |x|

otherwise. Show that

14^{-1} < g(x, t) < 14

x

t.

1

N sums are different [ILL 1977]

A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers

1, 2, 3, \ldots ,N

are marked, and on the ring

N

a_1,a_2,\ldots ,a_N

1

are marked. The ring can be turned into

N

different positions in which the numbers on the disk and on the ring match each other. Multiply every number on the ring with the corresponding number on the disk and form the sum of

N

products. In this way a sum is obtained for every position of the ring. Prove that the

N

sums are different.

1

Exists a lattice centroid point among 37 points [ILL 1977]

37

distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates.

1

E contains vertices of a tetrahedron - [ILL 1977]

E

be a finite set of points in space such that

E

is not contained in a plane and no three points of

E

are collinear. Show that

E

contains the vertices of a tetrahedron

T = ABCD

T \cap E = \{A,B,C,D\}

(including interior points of

T

) and such that the projection of

A

BCD

is inside a triangle that is similar to the triangle

BCD

and whose sides have midpoints

B,C,D.

1

Infinite set of solutions [ILL 1977]

A_1,A_2,\ldots ,A_{n+1}

be positive integers such that

(A_i,A_{n+1})=1

i=1,2,\ldots ,n

. Show that the equation

x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} }

has an infinite set of solutions

(x_1,x_2,\ldots , x_{n+1})

in positive integers.

1

All numbers in the sequence are different - [ILL 1977]

a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,

is defined by the following recurrence formula:

a_1 = 2,\qquad a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3

\text{for} k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots Prove that the numbers

a_{n,k}

are all different.

1

Mutliplying by 9 reverses digits [ILL 1977]

Find all numbers

N=\overline{a_1a_2\ldots a_n}

9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}

such that at most one of the digits

a_1,a_2,\ldots ,a_n

1

For every vector there is a scalar n [ILL 1977]

K

(0,0)

is given. Prove that for every vector

(a_1,a_2)

there is a positive integer

n

such that the circle

K

translated by the vector

n(a_1,a_2)

contains a lattice point (i.e., a point both of whose coordinates are integers).

1

Playing around with a two variable function [ILL 1977]

f

be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that

f

satisfies the following conditions:
(1)

f(ab,c)=f(a,c)f(b,c),\ f(c,ab)=f(c,a)f(c,b);

f(a,1-a)=1

f(a,a)=f(a,-a)=1,f(a,b)f(b,a)=1

1

Prove that there are at least three squares - [ILL 1977]

1, 2, 3,\ldots , 64

are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size

2 \times 2.

Prove that there are at least three such squares for which the sum of the

4

numbers contained exceeds

100.

1

Inequality for three sequences a_i, b_i and c_i - [ILL 1977]

m_j > 0

j = 1, 2,\ldots, n

a_1 \leq \cdots \leq a_n < b_1 \leq \cdots \leq b_n < c_1 \leq \cdots \leq c_n

be real numbers. Prove that

\Biggl( \sum_{j=1}^{n} m_j(a_j+b_j+c_j) \Biggr)^2 > 3 \Biggl( \sum_{j=1}^{n} m_j \Biggr) \Biggl( \sum_{j=1}^{n} m_j(a_jb_j+b_jc_j+c_ja_j) \Biggr).

1

Prove the binomial identity [ILL 1974]

Prove the identity

(z+a)^n=z^n+a\sum_{k=1}^n\binom{n}{k}(a-kb)^{k-1}(z+kb)^{n-k}

1

We will get the (1,1,...,1) sequence - [ILL 1977]

Consider a sequence of numbers

(a_1, a_2, \ldots , a_{2^n}).

Define the operation

S\biggl((a_1, a_2, \ldots , a_{2^n})\biggr) = (a_1a_2, a_2a_3, \ldots , a_{2^{n-1}a_{2^n}, a_{2^n}a_1).}

Prove that whatever the sequence

(a_1, a_2, \ldots , a_{2^n})

a_i \in \{-1, 1\}

i = 1, 2, \ldots , 2^n,

after finitely many applications of the operation we get the sequence

(1, 1, \ldots, 1).

1

Sum of x_i and y_i is 0 [ILL 1977]

x_1+x_2+x_3=y_1+y_2+y_3=x_1y_1+x_2y_2+x_3y_3=0,

\frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}=\frac{2}{3}

1

Last digits of 5^n are in a sequence [IL 1977]

Given any integer

m>1

prove that there exist infinitely many positive integers

n

such that the last

m

5^n

a_m,a_{m-1},\ldots ,a_1=5\ (0\le a_j<10)

in which each digit except the last is of opposite parity to its successor (i.e., if

a_i

a_{i-1}

a_i

a_{i-1}

1

Nine men in the room - [ILL 1977]

In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.

1

Prove that BS = BC (chosing the angles) - [ILL 1977]

ABC

\angle A = 30^\circ

\angle C = 54^\circ

BC

D

is chosen such that

\angle CAD = 12^\circ.

AB

E

is chosen such that

\angle ACE = 6^\circ.

S

be the point of intersection of

AD

CE.

BS = BC.

1

The tangent balls K,K_1,K_2 [ILL 1977]

K

r

is touched from the outside by mutually equal balls of radius

R

. Two of these balls are tangent to each other. Moreover, for two balls

K_1

K_2

K

and tangent to each other there exist two other balls tangent to

K_1,K_2

K

. How many balls are tangent to

K

r

R

1

Lattice points within the n x n first quadrant [ILL 1977]

n

be an integer greater than

1

. In the Cartesian coordinate system we consider all squares with integer vertices

(x,y)

1\le x,y\le n

p_k\ (k=0,1,2,\ldots )

the number of pairs of points that are vertices of exactly

k

such squares. Prove that

\sum_k(k-1)p_k=0

1

Indecomposable Numbers in the set V - [ILL 1977]

p

be a prime number greater than

5.

V

be the collection of all positive integers

n

that can be written in the form

n = kp + 1

n = kp - 1 \ (k = 1, 2, \ldots).

n \in V

is called indecomposable in

V

if it is impossible to find

k, l \in V

n = kl.

Prove that there exists a number

N \in V

that can be factorized into indecomposable factors in

V

in more than one way.

1

Find all functions (something like tangents) - [ILL 1971]

Determine all real functions

f(x)

that are defined and continuous on the interval

(-1, 1)

and that satisfy the functional equation

f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).

1

n must divide one of z_i [ILL 1977]

n

z

be integers greater than

1

(n,z)=1

. Prove: (a) At least one of the numbers

z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1,

is divisible by

n

(z-1,n)=1

, then at least one of the numbers

z_i

is divisible by

n

1

The non-empty set M and isometry Z

ABCD

be a regular tetrahedron and

\mathbf{Z}

an isometry mapping

A,B,C,D

B,C,D,A

, respectively. Find the set

M

X

ABC

whose distance from

\mathbf{Z}(X)

is equal to a given number

t

. Find necessary and sufficient conditions for the set

M

to be nonempty.

1

Construct the center and the radius of the circle-[ILL 1977]

Given an isosceles triangle

ABC

with a right angle at

C,

construct the center

M

r

of a circle cutting on segments

AB, BC, CA

DE, FG,

HK,

respectively, such that

\angle DME + \angle FMG + \angle HMK = 180^\circ

DE : FG : HK = AB : BC : CA.

1

Determine the set T - [ILL 1977]

z

> 1

M

be the set of all numbers of the form

z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots

. Determine the set

T

of divisors of at least one of the numbers

z_k

M.

1

Not quite Cauchy-Schwarz [ILL 1977]

Prove the following assertion: If

c_1,c_2,\ldots ,c_n\ (n\ge 2)

are real numbers such that

(n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,

then either all these numbers are nonnegative or all these numbers are nonpositive.

1

Find all points X for which Z(X) is minimal - [ILL 1977]

ABCDE

is made of two regular congruent tetrahedra

ABCD

ABCE.

Prove that there exists only one isometry

\mathbf Z

that maps points

A, B, C, D, E

B, C, A, E, D,

respectively. Find all points

X

on the surface of hexahedron whose distance from

\mathbf Z(X)

1

Inequaity with n people if k >= d [ILL 1977]

In a company of

n

persons, each person has no more than

d

acquaintances, and in that company there exists a group of

k

k\ge d

, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than

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

1

Sines and Cosines - [ILL 1977]

x_1, x_2, \ldots , x_n \ (n \geq 1)

be real numbers such that

0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.

\sum_{j=1}^n (\cos x_j +1)

is an odd integer, then

\sum_{j=1}^n \sin x_j \geq 1.

1

Pyramid with base on cylic pentagon

ABCDE

inscribed in a circle for which

BC<CD

AB<DE

is the base of a pyramid with vertex

S

AS

is the longest edge starting from

S

BS>CS

1

Maximal number of segments - [ILL 1977]

n

points in space. Some pairs of these points are connected by line segments so that the number of segments equals

[n^2/4],

and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle.