MathDB

1

Set of n points - ISL 1977

E

be a set of

n

points in the plane

(n \geq 3)

whose coordinates are integers such that any three points from

E

are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal

n.

14

1

A finite set of points - ISL 1977

Let

E

be a finite set of points such that

E

is not contained in a plane and no three points of

E

are collinear. Show that at least one of the following alternatives holds:
(i)

E

contains five points that are vertices of a convex pyramid having no other points in common with

E;

(ii) some plane contains exactly three points from

E.

13

1

At most k^2/2^n sequences

Let

B

be a set of

k

sequences each having

n

terms equal to

1

or

-1

. The product of two such sequences

(a_1, a_2, \ldots , a_n)

and

(b_1, b_2, \ldots , b_n)

is defined as

(a_1b_1, a_2b_2, \ldots , a_nb_n)

. Prove that there exists a sequence

(c_1, c_2, \ldots , c_n)

such that the intersection of

B

and the set containing all sequences from

B

multiplied by

(c_1, c_2, \ldots , c_n)

contains at most

\frac{k^2}{2^n}

sequences.

11

1

Prove that min(x1,x2,...,xn) = sqrt n

Let

n

be an integer greater than

1

. Define

x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,

where

[z]

denotes the largest integer less than or equal to

z

. Prove that

\min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]

9

1

Find n such that there exist two polynomials f, g

For which positive integers

n

do there exist two polynomials

f

and

g

with integer coefficients of

n

variables

x_1, x_2, \ldots , x_n

such that the following equality is satisfied:

\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ?

8

1

Prove that S4 is similar to S - ISL 1977

Let

S

be a convex quadrilateral

ABCD

and

O

a point inside it. The feet of the perpendiculars from

O

to

AB, BC, CD, DA

are

A_1, B_1, C_1, D_1

respectively. The feet of the perpendiculars from

O

to the sides of

S_i

, the quadrilateral

A_iB_iC_iD_i

, are

A_{i+1}B_{i+1}C_{i+1}D_{i+1}

, where

i = 1, 2, 3.

Prove that

S_4

is similar to S.

6

1

Integer solutions of the system of inequalities

Let

n

be a positive integer. How many integer solutions

(i, j, k, l) , \ 1 \leq i, j, k, l \leq n

, does the following system of inequalities have:

1 \leq -j + k + l \leq n

1 \leq i - k + l \leq n

1 \leq i - j + l \leq n

1 \leq i + j - k \leq n \ ?

5

1

2^n words - ISL 1987

There are

2^n

words of length

n

over the alphabet

\{0, 1\}

. Prove that the following algorithm generates the sequence

w_0, w_1, \ldots, w_{2^n-1}

of all these words such that any two consecutive words differ in exactly one digit.
(1)

w_0 = 00 \ldots 0

(

n

zeros).
(2) Suppose w_{m-1} = a_1a_2 \ldots a_n, a_i \in \{0, 1\}. Let

e(m)

be the exponent of

2

in the representation of

n

as a product of primes, and let

j = 1 + e(m)

. Replace the digit

a_j

in the word

w_{m-1}

by

1 - a_j

. The obtained word is

w_m

.

4

1

Describe all closed bounded figures Phi

Describe all closed bounded figures

\Phi

in the plane any two points of which are connectable by a semicircle lying in

\Phi

.

2

1

Lattice point and neighboring

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let

k

be a circle with radius

r \geq 2

, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle

k

that has a neighboring point lying outside

k

k

that has a neighboring point lying inside

k

. Prove that there are four more exterior boundary points than interior boundary points.

1

Function

Find all functions

f : \mathbb{N}\rightarrow \mathbb{N}

satisfying following condition: f(n+1)>f(f(n)), \forall n \in \mathbb{N}.

1977 IMO Shortlist

Subcontests

Set of n points - ISL 1977

A finite set of points - ISL 1977

At most k^2/2^n sequences

Prove that min(x1,x2,...,xn) = sqrt n

Find n such that there exist two polynomials f, g

Prove that S4 is similar to S - ISL 1977

Integer solutions of the system of inequalities

2^n words - ISL 1987

Describe all closed bounded figures Phi

Lattice point and neighboring

Function