2020 Junior Balkan Team Selection Tests-Serbia

Part of Serbia JBMO TST

Subcontests

(4)

1

Geometry

Given is triangle

ABC

with arbitrary point

D

AB

and central of inscribed circle

I

. The perpendicular bisector of

AB

AI

BI

P

Q

, respectively. The circle

(ADP)

CA

E

, and the circle

(BDQ)

BC

F

(ADP)

(BDQ)

K

E, F, K, I

lie on one circle.

1

Combinatorics

One hundred tennis players took part in a tournament where they played with each other exactly one game, with no draws. At the end of the tournament a table (ranking) is formed depending on the number of victories. It is known that one tennis player finished the tournament on

k

-th place and is the only one with that number of victories, and he has beaten every tennis player who is placed above him in the table and lost to anyone ranked weaker than him on the table. Find the smallest value of

k

1

Inequality

Given are real numbers

a_1, a_2,...,a_{101}

from the interval

[-2,10]

such that their sum is

0

. Prove that the sum of their squares is smaller than

2020

1

Number Theory

Solve in positive integers

x^{100}-y^{100}=100!