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Tangent to two circles
Mamadi 2
N
by A22-
Source: Own
Two circles 
and 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .
and 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .
2 replies
Deduction card battle
anantmudgal09 55
N
by deduck
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from 
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
Prove that the Detective can guarantee a win if and only if she is allowed to ask at least
questions.
Proposed by Anant Mudgal
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.Prove that the Detective can guarantee a win if and only if she is allowed to ask at least
questions.Proposed by Anant Mudgal
55 replies
1 viewing
Geometry
Lukariman 7
N
by vanstraelen
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that 
= 2∠AMP.
= 2∠AMP.
7 replies
perpendicularity involving ex and incenter
Erken 20
N
by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that 
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
20 replies
Isosceles Triangle Geo
oVlad 4
N
by Double07
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle 
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
4 replies
Geometry
Lukariman 1
N
by Primeniyazidayi
Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.
a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.
b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.
b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
1 reply
Kingdom of Anisotropy
v_Enhance 24
N
by deduck
Source: IMO Shortlist 2021 C4
The kingdom of Anisotropy consists of 
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.
Let
and 
be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from to . Similarly, let 
denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality 
holds if and only if the number of roads going out from is the same as the number of roads going out from .
Proposed by Warut Suksompong, Thailand
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.Let
and 
be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from to . Similarly, let 
denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality 
holds if and only if the number of roads going out from is the same as the number of roads going out from .Proposed by Warut Suksompong, Thailand
24 replies
Incentre-excentre geometry
oVlad 2
N
by Double07
Source: Romania Junior TST 2025 Day 2 P2
Consider a scalene triangle with incentre 
and excentres 
and 
, opposite the vertices 
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than .
with incentre 
and excentres 
and 
, opposite the vertices 
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than .2 replies
Great similarity
steven_zhang123 4
N
by khina
Source: a friend
As shown in the figure, there are two points 
and outside triangle such that 
and 
. Connect 
and 
, which intersect at point 
. Let 
intersect 
at point 
. Prove that 
.
and outside triangle such that 
and 
. Connect 
and 
, which intersect at point 
. Let 
intersect 
at point 
. Prove that 
.
4 replies
Unexpected FE
Taco12 18
N
by lpieleanu
Source: 2023 Fall TJ Proof TST, Problem 3
Find all functions 
such that for all integers 
and 
, ![\[ f(2x+f(y))+f(f(2x))=y. \]](//latex.artofproblemsolving.com/b/0/d/b0d16e87a8b30ebec5997ed12254094cdc3125d5.png)
Calvin Wang and Zani Xu
such that for all integers 
and 
, ![\[ f(2x+f(y))+f(f(2x))=y. \]](http://latex.artofproblemsolving.com/b/0/d/b0d16e87a8b30ebec5997ed12254094cdc3125d5.png)
Calvin Wang and Zani Xu
18 replies
