MathDB

1

Given two conditions related to x,y,z and min of a ratio wanted

x,y,z

be positive real numbers satisfying the equations

xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0

What is the minimum value of the ratio of the sum of the largest and smallest numbers among

x,y,z

to the median of them.

3

1

Solve a^7(a-1)=19b(19b+2) over Z

Find all pairs of integers

(a,b)

satisfying the equation

a^7(a-1)=19b(19b+2)

.

5

1

Interesting Configuration with Semicircle

We are given three points

A,B,C

on a semicircle. The tangent lines at

A

and

B

to the semicircle meet the extension of the diameter at points

M,N

respectively. The line passing through

A

that is perpendicular to the diameter meets

NC

at

R

, and the line passing through

B

that is perpendicular to the diameter meets

MC

at

S

. If the line

RS

meets the extension of the diameter at

Z

, prove that

ZC

is tangent to the semicircle.

4

1

Cute Combo with Process

On a table there are

100

red and

k

white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least once during the moves, the amount of water in these buckets is the same. Find all the possible values of

k

.

2

1

Combinatorics with Subsets

We are given some three element subsets of

\{1,2, \dots ,n\}

for which any two of them have at most one common element. We call a subset of

\{1,2, \dots ,n\}

nice if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element nice subset while keeping it nice, find the minimum value of

n

.

1

Reflections and circumcircle given, an angle equality wanted

Given an acute angle triangle

ABC

with circumcircle

\Gamma

and circumcenter

O

. A point

P

is taken on the line

BC

but not on

[BC]

. Let

K

be the reflection of the second intersection of the line

AP

and

\Gamma

with respect to

OP

. If

M

is the intersection of the lines

AK

and

OP

, prove that

\angle OMB+\angle OMC=180^{\circ}

.

2022 Turkey EGMO TST

Subcontests

Given two conditions related to x,y,z and min of a ratio wanted

Solve a^7(a-1)=19b(19b+2) over Z

Interesting Configuration with Semicircle

Cute Combo with Process

Combinatorics with Subsets

Reflections and circumcircle given, an angle equality wanted