MathDB

KM +ML = BC if AB = AC, <A = 30^o, AL = CM, <AMK = 45^o, <LMC = 75^&omicron;

Source: 2023 IGO Elementary P2

1/18/2024

In an isosceles triangle

ABC

with

AB = AC

and

\angle A = 30^o

, points

L

and

M

lie on the sides

AB

and

AC

, respectively such that

AL = CM

. Point

K

lies on

AB

such that

\angle AMK = 45^o

. If

\angle LMC = 75^o

, prove that

KM +ML = BC

.
Proposed by Mahdi Etesamifard - Iran

geometryequal segmentsisosceles

IGO 2023 Intermidiate P2

Source: IGO 2023 Intermidiate P2

1/18/2024

A convex hexagon

ABCDEF

with an interior point

P

is given. Assume that

BCEF

is a square and both

ABP

and

PCD

are right isosceles triangles with right angles at

B

and

C

, respectively. Lines

AF

and

DE

intersect at

G

. Prove that

GP

is perpendicular to

BC

.
Proposed by Patrik Bak - Slovakia

geometry

Why are equal areas so difficult?

Source: IGO 2023 Advanced P2

1/18/2024

Let

{I}

be the incenter of

\triangle {ABC}

and

{BX}

,

{CY}

are its two angle bisectors.

{M}

is the midpoint of arc

\overset{\frown}{BAC}

. It is known that

MXIY

are concyclic. Prove that the area of quadrilateral

MBIC

is equal to that of pentagon

BXIYC

.
Proposed by Dominik Burek - Poland

geometryincenterangle bisector

2

Problems(3)