MathDB

Similarity appearing from nowhere

Source: All-Russian 2021/9.4

4/19/2021

Given an acute triangle

ABC

, point

D

is chosen on the side

AB

and a point

E

is chosen on the extension of

BC

beyond

C

. It became known that the line through

E

parallel to

AB

is tangent to the circumcircle of

\triangle ADC

. Prove that one of the tangents from

E

to the circumcircle of

\triangle BCD

cuts the angle

\angle ABE

in such a way that a triangle similar to

\triangle ABC

is formed.

geometry

existence of some cards

Source: All-Russian 2021/10.4

4/19/2021

Given a natural number

n>4

and

2n+4

cards numbered with

1, 2, \dots, 2n+4

. On the card with number

m

a real number

a_m

is written such that

\lfloor a_{m}\rfloor=m

. Prove that it's possible to choose

4

cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than

\frac{1}{n-\sqrt{\frac{n}{2}}}

.

combinatoricsRussiaAll Russian Olympiad

angle bisectors and two congruent angles

Source: All-Russian 2021/11.4

4/20/2021

In triangle

ABC

angle bisectors

AA_{1}

and

CC_{1}

intersect at

I

. Line through

B

parallel to

AC

intersects rays

AA_{1}

and

CC_{1}

at points

A_{2}

and

C_{2}

respectively. Let

O_{a}

and

O_{c}

be the circumcenters of triangles

AC_{1}C_{2}

and

CA_{1}A_{2}

respectively. Prove that

\angle{O_{a}BO_{c}} = \angle{AIC}

geometryangle bisectorcircumcircle

4

Problems(3)