MathDB

1

Sums of numbers of the same colour not divisible by a given prime

p

and a positive integer

n

are given. Prove that one can colour every one of the numbers

1,2,\ldots,p-1

using one of the

2n

colours so that for any

i=2,3,\ldots,n

the sum of any

i

numbers of the same colour is not divisible by

p

.

5

1

An intersection of perpendicular bisectors and a weird angle condition

Let

ABC

be a triangle satisfying

AB<AC

. Let

M

be the midpoint of

BC

. A point

P

lies on the segment

AB

with

AP>PB

. A point

Q

lies on the segment

AC

with

\angle MPA = \angle AQM

. The perpendicular bisectors of

BC

and

PQ

intersect at

S

. Prove that

\angle BAC + \angle QSP = \angle QMP

.

4

1

Boring system of equations

Find all triples

(a,b,c)

of real numbers satisfying the system

\begin{cases} a^3+b^2c=ac \\ b^3+c^2a=ba \\ c^3+a^2b=cb \end{cases}

3

1

Pruning a highly connected graph

One has marked

n

points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any

2021

drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most

2022n

chords remain and the property described above is preserved.

2

1

Fractions divisible by a fixed integer

Let

m,n\ge 2

be given integers. Prove that there exist positive integers

a_1<a_2<\ldots<a_m

so that for any

1\le i<j\le m

the number

\frac{a_j}{a_j-a_i}

is an integer divisible by

n

.

1

Lots of perpendiculars

Let

ABC

be an acute triangle with

AB<AC

. The angle bisector of

BAC

intersects the side

BC

and the circumcircle of

ABC

at

D

and

M\neq A

, respectively. Points

X

and

Y

are chosen so that

MX \perp AB

,

BX \perp MB

,

MY \perp AC

, and

CY \perp MC

. Prove that the points

X,D,Y

are collinear.

2022 Polish MO Finals

Subcontests

Sums of numbers of the same colour not divisible by a given prime

An intersection of perpendicular bisectors and a weird angle condition

Boring system of equations

Pruning a highly connected graph

Fractions divisible by a fixed integer

Lots of perpendiculars