2

Part of 2022 Tuymaada Olympiad

Problems(2)

Source: Tuymaada 2022 Junior P-2

Given are integers

a, b, c

and an odd prime

p.

p

x^2 + y^2 + ax + by + c

for some integers

x

y.

(A. Golovanov )

number theorypolynomial

Tuymaada P2 senior

Source: tuymaada 2022 senior P2

w_{1}

w_{2}

of different radii touch externally at

L

. A line touches

w_{1}

A

w_{2}

B

A

B

are different from

L

X

is chosen in the plane.

Y

Z

are the second points of intersection of the lines

XA

XB

w_{1}

w_{2}

respectively. Prove that all

X

AB||Y Z

belong to one circle.