P3

Part of 2020/2021 Tournament of Towns

Problems(6)

Incenter coincides with intersection of diagonals

Source: 42nd International Tournament of Towns, Senior A-Level P3, Fall 2020

2/18/2023

\alpha{}

and

\beta{}

with centers

A{}

and

B{}

respectively intersect at points

C{}

and

D{}

. The segment

AB{}

intersects

\alpha{}

and

\beta{}

at points

K{}

and

L{}

respectively. The ray

DK

intersects the circle

\beta{}

for the second time at the point

N{}

, and the ray

DL

intersects the circle

\alpha{}

for the second time at the point

M{}

. Prove that the intersection point of the diagonals of the quadrangle

KLMN

coincides with the incenter of the triangle

ABC

.
Konstantin Knop

geometryTournament of Towns

Bob writes numbers according to Alice's instructions

Source: 42nd International Tournament of Towns, Junior A-Level P3, Fall 2020

2/18/2023

Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are
[*]at least a hundred copies of number 5? [*]at least a hundred copies of number 10?
Andrey Arzhantsev

gamecombinatoricsTournament of Towns

Swapping digits cancels divisibility

Source: 42nd International Tournament of Towns, Senior O-Level P3, Fall 2020

2/18/2023

A positive integer number

N{}

is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number

N{}

have?
Sergey Tokarev

number theoryDivisibilityTournament of Towns

Game with stones in a heap

Source: 42nd International Tournament of Towns, Junior O-Level P3, Fall 2020

2/18/2023

There are

n{}

stones in a heap. Two players play the game by alternatively taking either 1 stone from the heap or a prime number of stones which divides the current number of stones in the heap. The player who takes the last stone wins. For which

n{}

does the first player have a strategy so that he wins no matter how the other player plays?
Fedor Ivlev

combinatoricsgameTournament of Towns

Tangent circles geo

Source: 42nd International Tournament of Towns, Senior A-Level P3, Spring 2021

2/18/2023

Let

M{}

be the midpoint of the side

BC

of the triangle

ABC

. The circle

\omega

passes through

A{}

, touches the line

BC

M{}

, intersects the side

AB

at the point

D{}

and the side

AC

at the point

E{}

. Let

X{}

and

Y{}

be the midpoints of

BE

and

CD

respectively. Prove that the circumcircle of the triangle

MXY

touches

\omega

.
Alexey Doledenok

geometrycirclesTournament of Towns

The angle KPC is right

Source: 42nd International Tournament of Towns, Junior A-Level P3, Spring 2021

2/18/2023

There is an equilateral triangle

ABC

. Let

E, F

and

K

be points such that

E{}

lies on side

AB

F{}

lies on the side

AC

K{}

lies on the extension of side

AB

and

AE = CF = BK

. Let

P{}

be the midpoint of the segment

EF

. Prove that the angle

KPC

is right.
Vladimir Rastorguev

geometryTournament of Towns