G
Topic
First Poster
Last Poster
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
Geometry
Lukariman 6
N
by Curious_Droid
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.
6 replies
Powers of a Prime
numbertheorist17 33
N
by OronSH
Source: USA TSTST 2014, Problem 6
Suppose we have distinct positive integers 
, and an odd prime 
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
, and an odd prime 
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
33 replies
Expected Intersections from Random Pairing on a Circle
tom-nowy 2
N
by lele0305
Let 
be a positive integer. Consider 
points on the circumference of a circle.
These points are randomly divided into pairs, and line segments are drawn connecting the points in each pair.
Find the expected number of intersection points formed by these segments, assuming no three segments intersect at a single point.
be a positive integer. Consider 
points on the circumference of a circle.These points are randomly divided into
pairs, and line segments are drawn connecting the points in each pair.Find the expected number of intersection points formed by these segments, assuming no three segments intersect at a single point.
2 replies
question4
sahadian 5
N
by Mamadi
Source: iran tst 2014 first exam
Find the maximum number of Permutation of set {
} such that for every 2 different number 
and 
in this set at last in one of the permutation
comes exactly after
} such that for every 2 different number 
and 
in this set at last in one of the permutation comes exactly after
5 replies
Find all functions $f$: \(\mathbb{R^+}\) \(\rightarrow\) \(\mathbb{R^+}\) such
guramuta 5
N
by jasperE3
Source: Balkan MO SL 2021
A5: Find all functions 
: 
such that:
: 
such that:
5 replies
number theory
frost23 3
N
by frost23
given any positive integer n show that there are two positive rational numbers a and b not equal to b which are such that a-b, a^2- b^2....................a^n-b^n are all integers
3 replies
partitioned square
moldovan 8
N
by cursed_tangent1434
Source: Ireland 1994
If a square is partitioned into 
convex polygons, determine the maximum possible number of edges in the obtained figure.
(You may wish to use the following theorem of Euler: If a polygon is partitioned into polygons with 
vertices and 
edges in the resulting figure, then 
.)
convex polygons, determine the maximum possible number of edges in the obtained figure.(You may wish to use the following theorem of Euler: If a polygon is partitioned into
polygons with 
vertices and 
edges in the resulting figure, then 
.)
8 replies
