Y by GeoKing, parmenides51, Funcshun840, NicoN9, L13832
If there’s any typo or problem you think will be a nice addition, do send here!
handout, geometry
handout, geometry
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equal segments on radiuses
danepale 8
N
by zuat.e
Source: Croatia TST 2016
Let 
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
8 replies
2- player game on a strip of n squares with two game pieces
parmenides51 2
N
by Gggvds1
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 3
Alice and Bob play a game on a strip of 
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares.
IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which
can Bob ensure a win no matter how Alice plays?
For which can Alice ensure a win no matter how Bob plays?
(Karl Czakler)
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares.IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which
can Bob ensure a win no matter how Alice plays?For which
can Alice ensure a win no matter how Bob plays?(Karl Czakler)
2 replies
Incenters and Circles
rkm0959 6
N
by happypi31415
Source: Korean National Junior Olympiad Problem 1
In a triangle 
with incenter 
,
Let
= 
= incenter of 
= incenter of 
= intersection of 
and 
= intersection of 
and 
= midpoint of 
Prove that
concycle
with incenter 
,Let
= 
= incenter of 
= incenter of 
= intersection of 
and 
= intersection of 
and 
= midpoint of 
Prove that
concycle
6 replies
Reflected point lies on radical axis
Mahdi_Mashayekhi 6
N
by khanhnx
Source: Iran 2025 second round P4
Given is an acute and scalene triangle with circumcenter . 
and 
intersect the altitude from 
to at points and respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
with circumcenter . 
and 
intersect the altitude from 
to at points and respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
6 replies
Gcd of N and its coprime pair sum
EeEeRUT 19
N
by HamstPan38825
Source: EGMO 2025 P1
For a positive integer , let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
Here
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
Proposed by Paulius Aleknavičius, Lithuania
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
Here
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.Proposed by Paulius Aleknavičius, Lithuania
19 replies
Question on Balkan SL
Fmimch 4
N
by BreezeCrowd
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
4 replies
(3^{p-1} - 1)/p is a perfect square for prime p
parmenides51 4
N
by Rayvhs
Source: 2017 Saudi Arabia JBMO TST 1.2
Find all prime numbers 
such that 
is a perfect square.
such that 
is a perfect square.
4 replies
Rays, incircle, angles...
mathisreal 3
N
by Assassino9931
Source: Rioplatense L-3 2022 #4
Let be a triangle with incenter . Let be the point of intersection between the incircle and the side , the points and are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
be a triangle with incenter . Let be the point of intersection between the incircle and the side , the points and are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
3 replies
Find the value
sqing 0
Source: Own
Let 
be real numbers such that 
. Find the value of 
Let be real numbers such that 
and 
. Find the value of 
be real numbers such that 
. Find the value of 
Let
be real numbers such that 
and 
. Find the value of 
0 replies
Wordy Geometry in Taiwan TST
ckliao914 9
N
by Scilyse
Source: 2023 Taiwan TST Round 3 Mock Exam 6
Given triangle with -excenter 
, the foot of the perpendicular from to is . Let the midpoint of segment 
be , 
lies on arc (not containing ) satisfying 
, 
intersects the circumcircle of at 
. If 
and intersect at , the perpendicular bisector of 
intersects 
at 
respectively, prove that 
are concurrent.
with -excenter 
, the foot of the perpendicular from to is . Let the midpoint of segment 
be , 
lies on arc (not containing ) satisfying 
, 
intersects the circumcircle of at 
. If 
and intersect at , the perpendicular bisector of 
intersects 
at 
respectively, prove that 
are concurrent.
9 replies
Factorial Divisibility
Aryan-23 47
N
by ezpotd
Source: IMO SL 2022 N2
Find all positive integers 
such that
such that
47 replies
Geometry Handout is finally done!
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Y by Rounak_iitr, AshAuktober
posts like this remind me that a geo handout collection should be made, in order those worthy posts to be collected at one place
someday, I will create one
someday, I will create one
This post has been edited 1 time. Last edited by parmenides51, Apr 25, 2025, 12:30 AM
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