MathDB

Regional Olympiad - FBH 2016 Grade 9 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018

Nine lines are given such that every one of them intersects given square

ABCD

on two trapezoids, which area ratio is

2 : 3

. Prove that at least

3

of those

9

lines pass through the same point

geometrytrapezoidratio

Regional Olympiad - FBH 2016 Grade 10 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018

Let

AB

be a diameter of semicircle

h

. On this semicircle there is point

C

, distinct from points

A

and

B

. Foot of perpendicular from point

C

to side

AB

is point

D

. Circle

k

is outside the triangle

ADC

and at the same time touches semicircle

h

and sides

AB

and

CD

. Touching point of

k

with side

AB

is point

E

, with semicircle

h

is point

T

and with side

CD

is point

S

a)

Prove that points

A

,

S

and

T

are collinear

b)

Prove that

AC=AE

geometrycollinearsemicircletouching circles

Regional Olympiad - FBH 2016 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018

h_a

,

h_b

and

h_c

are altitudes,

t_a

,

t_b

and

t_c

are medians of acute triangle,

r

radius of incircle, and

R

radius of circumcircle of acute triangle

ABC

. Prove that

\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}

geometrycircumcircle

Regional Olympiad - FBH 2016 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018

Circle of radius

R_1

is inscribed in an acute angle

\alpha

. Second circle with radius

R_2

touches one of the sides forming the angle

\alpha

in same point as first circle and intersects the second side in points

A

and

B

, such that centers of both circles lie inside angle

\alpha

. Prove that

AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}

circlegeometry

3

Problems(4)