MathDB

AD + DC>= AB if CD = BC = BE and ABCD is convex with E =AC\cap BD

Source: 2023 IGO Elementary P4

1/18/2024

Let

ABCD

be a convex quadrilateral. Let

E

be the intersection of its diagonals. Suppose that

CD = BC = BE

. Prove that

AD + DC\ge AB

.
Proposed by Dominik Burek - Poland

geometrygeometric inequality

AB, SH , TR concurrent wanted , HAPQ and SACQ are #

Source: 2023 IGO Intermediate P4

1/18/2024

Let

ABC

be a triangle and

P

be the midpoint of arc

BAC

of circumcircle of triangle

ABC

with orthocenter

H

. Let

Q, S

be points such that

HAPQ

and

SACQ

are parallelograms. Let

T

be the midpoint of

AQ

, and

R

be the intersection point of the lines

SQ

and

PB

. Prove that

AB

,

SH

and

TR

are concurrent.
Proposed by Dominik Burek - Poland

geometryconcurrencyconcurrent

line AI bisects segment XY , XY _|_ IP

Source: 2023 IGO Adcanced P4

1/18/2024

Let

ABC

be a triangle with bisectors

BE

and

CF

meet at

I

. Let

D

be the projection of

I

on the

BC

. Let M and

N

be the orthocenters of triangles

AIF

and

AIE

, respectively. Lines

EM

and

FN

meet at

P.

Let

X

be the midpoint of

BC

. Let

Y

be the point lying on the line

AD

such that

XY \perp IP

. Prove that line

AI

bisects the segment

XY

.
Proposed by Tran Quang Hung - Vietnam

geometrybisects segment

4

Problems(3)