MathDB

1

Can you tell the integer written by the other student?

A

and

B

are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student

A:

“Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student

B.

If

B

answers “no,” the referee puts the question back to

A,

and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

29

1

Superinvariant partial affine transformations

We call a set

S

on the real line

\mathbb{R}

superinvariant if for any stretching

A

of the set by the transformation taking

x

to A(x) \equal{} x_0 \plus{} a(x \minus{} x_0), a > 0 there exists a translation

B,

B(x) \equal{} x\plus{}b, such that the images of

S

under

A

and

B

agree; i.e., for any

x \in S

there is a

y \in S

such that A(x) \equal{} B(y) and for any

t \in S

there is a

u \in S

such that B(t) \equal{} A(u). Determine all superinvariant sets.

27

1

Maximum value of x_ix_j (x_i + x_j) summed over all i

Determine the maximum value of the sum \sum_{i < j} x_ix_j (x_i \plus{} x_j) over all n \minus{}tuples

(x_1, \ldots, x_n),

satisfying

x_i \geq 0

and \sum^n_{i \equal{} 1} x_i \equal{} 1.

26

1

Inequality for geometric and arithmetic mean for n-1 numbers

Let

n \geq 2, n \in \mathbb{N}

and let

p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}

satisfying

\frac{1}{2} \leq p \leq 1,

0 \leq a_i,

0 \leq b_i \leq p,

i \equal{} 1, \ldots, n, and \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i. Prove the inequality: \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.

25

1

x_i+1 <= 0.5

Suppose that

n \geq 2

and

x_1, x_2, \ldots, x_n

are real numbers between 0 and 1 (inclusive). Prove that for some index

i

between

1

and n \minus{} 1 the inequality x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})

23

1

f(m + f(f(n))) = -f(f(m+ 1) - n

Let

f

and

g

be two integer-valued functions defined on the set of all integers such that (a) f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n for all integers

m

and

n;

(b)

g

is a polynomial function with integer coefficients and g(n) =

g(f(n))

\forall n \in \mathbb{Z}.

22

1

Vertices lie on the cubic curve y = x^3 + ax^2 + bx + c

Real constants

a, b, c

are such that there is exactly one square all of whose vertices lie on the cubic curve y \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c. Prove that the square has sides of length

\sqrt[4]{72}.

21

1

Monic polynomial of degree 1991 with integer coefficients

Let

f(x)

be a monic polynomial of degree

1991

with integer coefficients. Define g(x) \equal{} f^2(x) \minus{} 9. Show that the number of distinct integer solutions of g(x) \equal{} 0 cannot exceed

1995.

18

1

Highest degree for 3-layer power tower (IMO ShortList 1991)

Find the highest degree

k

of

1991

for which

1991^k

divides the number 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.

17

1

3^x + 4^y = 5^z

Find all positive integer solutions

x, y, z

of the equation 3^x \plus{} 4^y \equal{} 5^z.

15

1

Does sequence become periodic after finite number of terms?

Let

a_n

be the last nonzero digit in the decimal representation of the number

n!.

Does the sequence

a_1, a_2, \ldots, a_n, \ldots

become periodic after a finite number of terms?

14

1

f(x) = ax^2 + bx + c is a perfect square for 2p - 1 integers

Let

a, b, c

be integers and

p

an odd prime number. Prove that if f(x) \equal{} ax^2 \plus{} bx \plus{} c is a perfect square for 2p \minus{} 1 consecutive integer values of

x,

then

p

divides b^2 \minus{} 4ac.

13

1

Dot product divisible by n

Given any integer

n \geq 2,

assume that the integers

a_1, a_2, \ldots, a_n

are not divisible by

n

and, moreover, that

n

does not divide \sum^n_{i\equal{}1} a_i. Prove that there exist at least

n

different sequences

(e_1, e_2, \ldots, e_n)

consisting of zeros or ones such \sum^n_{i\equal{}1} e_i \cdot a_i is divisible by

n.

9

1

At least 1593 points of E to which it is joined by a path

In the plane we are given a set

E

of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of

E,

there exist at least 1593 other points of

E

to which it is joined by a path. Show that there exist six points of

E

every pair of which are joined by a path. Alternative version: Is it possible to find a set

E

of 1991 points in the plane and paths joining certain pairs of the points in

E

such that every point of

E

is joined with a path to at least 1592 other points of

E,

and in every subset of six points of

E

there exist at least two points that are not joined?

8

1

Every triangle whose three vertices are elements of S

S

be a set of

n

points in the plane. No three points of

S

are collinear. Prove that there exists a set

P

containing 2n \minus{} 5 points satisfying the following condition: In the interior of every triangle whose three vertices are elements of

S

lies a point that is an element of

P.

5

1

Drawn through the incenter I of the triangle

In the triangle

ABC,

with \angle A \equal{} 60 ^{\circ}, a parallel

IF

to

AC

is drawn through the incenter

I

of the triangle, where

F

lies on the side

AB.

The point

P

on the side

BC

is such that 3BP \equal{} BC. Show that \angle BFP \equal{} \frac{\angle B}{2}.

3

1

Feet of perpendiculars from S, Simson Line

Let

S

be any point on the circumscribed circle of

PQR.

Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by

l(S, PQR).

Suppose that the hexagon

ABCDEF

is inscribed in a circle. Show that the four lines

l(A,BDF),

l(B,ACE),

l(D,ABF),

and

l(E,ABC)

intersect at one point if and only if

CDEF

is a rectangle.

24

1

O 32

An odd integer

n \ge 3

is said to be nice if and only if there is at least one permutation

a_{1}, \cdots, a_{n}

of

1, \cdots, n

such that the

n

sums a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}, a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}, a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2},

\cdots

, a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1} are all positive. Determine the set of all `nice' integers.

20

1

I 1

Let

\alpha

be the positive root of the equation x^{2} \equal{} 1991x \plus{} 1. For natural numbers

m

and

n

define m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. Prove that for all natural numbers

p

,

q

, and

r

, (p*q)*r \equal{} p*(q*r).

19

1

F 3

Let

\alpha

be a rational number with

0 < \alpha < 1

and \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0. Prove that \alpha \equal{} \frac {2}{3}.

2

1

Another geometry problem :)

ABC

is an acute-angled triangle.

M

is the midpoint of

BC

and

P

is the point on

AM

such that MB \equal{} MP.

H

is the foot of the perpendicular from

P

to

BC

. The lines through

H

perpendicular to

PB

,

PC

meet

AB, AC

respectively at

Q, R

. Show that

BC

is tangent to the circle through

Q, H, R

at

H

. Original Formulation: For an acute triangle

ABC, M

is the midpoint of the segment

BC, P

is a point on the segment

AM

such that PM \equal{} BM, H is the foot of the perpendicular line from

P

to

BC, Q

is the point of intersection of segment

AB

and the line passing through

H

that is perpendicular to

PB,

and finally,

R

is the point of intersection of the segment

AC

and the line passing through

H

that is perpendicular to

PC.

Show that the circumcircle of

QHR

is tangent to the side

BC

at point

H.

11

1

combinatorial sum

Prove that \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}

1

lacsap

Given a point

P

inside a triangle

\triangle ABC

. Let

D

,

E

,

F

be the orthogonal projections of the point

P

on the sides

BC

,

CA

,

AB

, respectively. Let the orthogonal projections of the point

A

on the lines

BP

and

CP

be

M

and

N

, respectively. Prove that the lines

ME

,

NF

,

BC

are concurrent. Original formulation: Let

ABC

be any triangle and

P

any point in its interior. Let

P_1, P_2

be the feet of the perpendiculars from

P

to the two sides

AC

and

BC.

Draw

AP

and

BP,

and from

C

drop perpendiculars to

AP

and

BP.

Let

Q_1

and

Q_2

be the feet of these perpendiculars. Prove that the lines

Q_1P_2,Q_2P_1,

and

AB

are concurrent.

7

1

tetrahedron

ABCD

is a terahedron: AD\plus{}BD\equal{}AC\plus{}BC, BD\plus{}CD\equal{}BA\plus{}CA, CD\plus{}AD\equal{}CB\plus{}AB,

M,N,P

are the mid points of

BC,CA,AB.

OA\equal{}OB\equal{}OC\equal{}OD. Prove that \angle MOP \equal{} \angle NOP \equal{}\angle NOM.

1991 IMO Shortlist

Subcontests

Can you tell the integer written by the other student?

Superinvariant partial affine transformations

Maximum value of x_ix_j (x_i + x_j) summed over all i

Inequality for geometric and arithmetic mean for n-1 numbers

x_i+1 <= 0.5

f(m + f(f(n))) = -f(f(m+ 1) - n

Vertices lie on the cubic curve y = x^3 + ax^2 + bx + c

Monic polynomial of degree 1991 with integer coefficients

Highest degree for 3-layer power tower (IMO ShortList 1991)

3^x + 4^y = 5^z

Does sequence become periodic after finite number of terms?

f(x) = ax^2 + bx + c is a perfect square for 2p - 1 integers

Dot product divisible by n

At least 1593 points of E to which it is joined by a path

Every triangle whose three vertices are elements of S

Drawn through the incenter I of the triangle

Feet of perpendiculars from S, Simson Line

O 32

I 1

F 3

Another geometry problem :)

combinatorial sum

lacsap

tetrahedron

1991 IMO Shortlist

Subcontests

Can you tell the integer written by the other student?

Superinvariant partial affine transformations

Maximum value of x_i*x_j* (x_i + x_j) summed over all i

Inequality for geometric and arithmetic mean for n-1 numbers

x_i+1 &lt;= 0.5

f(m + f(f(n))) = -f(f(m+ 1) - n

Vertices lie on the cubic curve y = x^3 + ax^2 + bx + c

Monic polynomial of degree 1991 with integer coefficients

Highest degree for 3-layer power tower (IMO ShortList 1991)

3^x + 4^y = 5^z

Does sequence become periodic after finite number of terms?

f(x) = ax^2 + bx + c is a perfect square for 2p - 1 integers

Dot product divisible by n

At least 1593 points of E to which it is joined by a path

Every triangle whose three vertices are elements of S

Drawn through the incenter I of the triangle

Feet of perpendiculars from S, Simson Line

O 32

I 1

F 3

Another geometry problem :)

combinatorial sum

lacsap

tetrahedron

Maximum value of x_ix_j (x_i + x_j) summed over all i

x_i+1 <= 0.5