MathDB

Problem 1 of Second round

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/22/2020

Let

u_1=1,u_2=2,u_3=24,

and

u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.

Prove that the elements of the sequence are natural numbers and that

n\mid u_n

for all

n

.

algebraDivisibilitySequence

Geometry

Source: Own

9/21/2019

Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE intersects BD at F. It is known that triangle BEF is equilateral. Find

geometry

Problem 1 of Fourth round

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/24/2020

Let point

T

be on side

AB

of

\Delta ABC

be such that

AT-BT=AC-BC

. The perpendicular from point

T

to

AB

intersects

AC

in point

E

and the angle bisectors of

\angle B

and

\angle C

intersect the circumscribed circle

k

of

ABC

in points

M

and

L

. If

P

is the second intersection point of the line

ME

with

k

, then prove that

P,T,L

are collinear.

geometrycollinearPascal s theorem

Problem 1 of Finals

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/25/2020

The points

P

and

Q

on the side

AC

of the non-isosceles

\Delta ABC

are such that

\angle ABP=\angle QBC<\frac{1}{2}\angle ABC

. The angle bisectors of

\angle A

and

\angle C

intersect the segment

BP

in points

K

and

L

and the segment

BQ

in points

M

and

N

, respectively. Prove that

AC

,

KN

, and

LM

are concurrent.

geometryconcurrencymoving points

1

Problems(4)