MathDB

Collinearity in Isosceles Trapezoid

Source: All Russian Olympiad 2017 Grade 9 Problem 2

7/5/2017

ABCD

is an isosceles trapezoid with

BC || AD

. A circle

\omega

passing through

B

and

C

intersects the side

AB

and the diagonal

BD

at points

X

and

Y

respectively. Tangent to

\omega

at

C

intersects the line

AD

at

Z

. Prove that the points

X

,

Y

, and

Z

are collinear.

geometrytrapezoidtangent

Integer polynomial

Source: All Russian Olympiad 2017,Day2,grade 9,P6

5/3/2017

a,b,c

- different natural numbers. Can we build quadratic polynomial

P(x)=kx^2+lx+m

, with

k,l,m

are integer,

k>0

that for some integer points it get values

a^3,b^3,c^3

?

quadraticsnumber theoryalgebrapolynomial

Tangent in Isosceles Triangle

Source: All Russian Olympiad 2017 10.2 & 11.2

7/5/2017

Let

ABC

be an acute angled isosceles triangle with

AB=AC

and circumcentre

O

. Lines

BO

and

CO

intersect

AC, AB

respectively at

B', C'

. A straight line

l

is drawn through

C'

parallel to

AC

. Prove that the line

l

is tangent to the circumcircle of

\triangle B'OC

.

tangentgeometryisoscelescircumcircle

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