Y by
Let 
be natural numbers such that: 
and 
. Find the min: 
be natural numbers such that: 
and 
. Find the min: 
G
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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
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
Minimum with natural numbers
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Y by
We have
so is odd and is even. Say 
, with 
, then
so 
or 
.
If then
so 
, and only 
gives a solution in natural numbers. In this case we get 
, which gives 
, so this doesn't work.
Hence, and so 
. Then
so 
, and only 
gives a solution in natural numbers. In this case we get 
, which gives 
, so this doesn't work.
Hence I assume you mean whole numbers rather than natural numbers.
In this case gives the solution 
and 
.
We also have to check the case
. Here we have
so 
, and only 
gives a solution in whole numbers. In this case we get 
, which gives 
, so this doesn't work.
Hence there are no solutions in natural numbers and only one solution in whole numbers.
so is odd and is even. Say 
, with 
, then
so 
or 
.If
then
so 
, and only 
gives a solution in natural numbers. In this case we get 
, which gives 
, so this doesn't work.Hence
, and so 
. Then
so 
, and only 
gives a solution in natural numbers. In this case we get 
, which gives 
, so this doesn't work.Hence I assume you mean whole numbers rather than natural numbers.
In this case
gives the solution 
and 
.We also have to check the case
. Here we have
so 
, and only 
gives a solution in whole numbers. In this case we get 
, which gives 
, so this doesn't work.Hence there are no solutions in natural numbers and only one solution in whole numbers.
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K
Y
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G
H
