MathDB

1

Find the angle

ABC

be an acute triangle. Let

P

be a point on the circle

(ABC)

, and

Q

be a point on the segment

AC

such that

AP\perp BC

and

BQ\perp AC

. Lot

O

be the circumcenter of triangle

APQ

. Find the angle

OBC

.

4

1

Intersection point on circle

Let

\omega

is tangent to the sides of an acute angle with vertex

A

at points

B

and

C

. Let

D

be an arbitrary point onn the major arc

BC

of the circle

\omega

. Points

E

and

F

are chosen inside the angle

DAC

so that quadrilaterals

ABDF

and

ACED

are inscribed and the points

A,E,F

lie on the same straight line. Prove that lines

BE

and

CF

intersectat

\omega

.

8

1

Upper bound for reciprocals of binomial coefficients

Paul can write polynomial

(x+1)^n

, expand and simplify it, and after that change every coefficient by its reciprocal. For example if

n=3

Paul gets

(x+1)^3=x^3+3x^2+3x+1

and then

x^3+\frac13x^2+\frac13x+1

. Prove that Paul can choose

n

for which the sum of Paul’s polynomial coefficients is less than

2.022

.

7

1

Variable line touches a fixed circle

Point

P

is chosen on the leg

CB

of right triangle

ABC

(

\angle ACB = 90^\circ

). The line

AP

intersects the circumcircle of

ABC

at point

Q

. Let

L

be the midpoint of

PB

. Prove that

QL

is tangent to a fixed circle independent of the choice of point

P

.

5

1

Diophantine equation in integers with odd digits

Let

S

be the set of all

5^6

positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions

(x,y,z)

of the equation

x+y=10z

, where

x\in S

,

y\in S

,

z\in S

.

3

1

Set of numbers and function a+b-gcd(a;b)

Pete wrote down

21

pairwise distinct positive integers, each not greater than

1,000,000

. For every pair

(a, b)

of numbers written down by Pete, Nick wrote the number

F(a;b)=a+b -\gcd(a;b)

on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.

2

1

Easy similarity in parallelogram

In parallelogram

ABCD

, points

E

and

F

on segments

AD

and

CD

are such that

\angle BCE=\angle BAF

. Points

K

and

L

on segments

AD

and

CD

are such that

AK=ED

and

CL=FD

. Prove that

\angle BKD=\angle BLD

.

1

2

Double of a squares in geometric progression

Positive integers

a

,

b

,

c

are given. It is known that

\frac{c}{b}=\frac{b}{a}

, and the number

b^2-a-c+1

is a prime. Prove that

a

and

c

are double of a squares of positive integers.

In each column different numbers but fixed sum in second row

Given a rectangular table with 2 rows and 100 columns. Dima fills the cells of the first row with numbers 1, 2 or 3. Prove that Alex can fill the cells of the second row with numbers 1, 2, 3 in such a way that the numbers in each column are different and the sum of the numbers in the second row equals 200.

2022 Caucasus Mathematical Olympiad

Subcontests

Find the angle

Intersection point on circle

Upper bound for reciprocals of binomial coefficients

Variable line touches a fixed circle

Diophantine equation in integers with odd digits

Set of numbers and function a+b-gcd(a;b)

Easy similarity in parallelogram

Double of a squares in geometric progression

In each column different numbers but fixed sum in second row