MathDB

Regional Olympiad - FBH 2012 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018

Let

S

be an incenter of triangle

ABC

and let incircle touch sides

AC

and

AB

in points

P

and

Q

, respectively. Lines

BS

and

CS

intersect line

PQ

in points

M

and

N

, respectively. Prove that points

M

,

N

,

B

and

C

are concyclic

geometryincenterincircle

Regional Olympiad - FBH 2012 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018

Can number

2012^n-3^n

be perfect square, while

n

is positive integer

number theoryPerfect Square

Regional Olympiad - FBH 2012 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018

Prove the inequality:

\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}

where

A

,

B

,

C

,

a

,

b

,

c

and

r

are positive real numbers

inequalitiesalgebra

Regional Olympiad - FBH 2012 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018

In triangle

ABC

point

O

is circumcenter. Point

T

is centroid of

ABC

, and points

D

,

E

and

F

are circumcenters of triangles

TBC

,

TCA

and

TAB

. Prove that

O

is centroid of

DEF

geometrycircumcircle

4

Problems(4)