2021 Polish Junior MO Finals

Part of 2021 Polish Junior Math Olympiad

Subcontests

(5)

1

numbers with the same digits

Natural numbers

a

b

are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in

a

b

). Prove that if

a+b=10^{1000}

then both numbers

a

b

are divisible by

10

1

angle equality in a scalene triangle

AB

of a scalene triangle

ABC

there are points

M

N

AN=AC

BM=BC

. The line parallel to

BC

M

and the line parallel to

AC

N

S

\measuredangle{CSM} = \measuredangle{CSN}

1

tournament, different numbers of wins imply different numbers of losses

In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games. Prove that each participant has lost a different number of games.

1

An inequality in right angled triangle

M

is the midpoint of the hypotenuse

AB

of a right angled triangle

ABC

P

Q

lie on segments

AM

MB

respectively and

PQ=CQ

AP\leq 2\cdot MQ

1

Integer square root given fraction identity

Positive integers

a

b

n

\frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}.

\sqrt{ab}