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P2 Geo that most of contestants died
AlephG_64 2
N
by Tsikaloudakis
Source: 2025 Finals Portuguese Mathematical Olympiad P2
Let 
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
2 replies
Geometry
youochange 0
m:}
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
.
Prove that the point lies on the circumcircle of 
.
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
.Prove that the point
lies on the circumcircle of 
.
0 replies
comp. geo starting with a 90-75-15 triangle. <APB =<CPQ, <BQA =<CQP.
parmenides51 1
N
by Mathzeus1024
Source: 2013 Cuba 2.9
Let ABC be a triangle with 
, 
, and 
. Points and of the sides and respectively, are such that 
and 
. Calculate the lenght of 
.
, 
, and 
. Points and of the sides and respectively, are such that 
and 
. Calculate the lenght of 
.
1 reply
Fridolin just can't get enough from jumping on the number line
Tintarn 2
N
by Sadigly
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 1
Fridolin the frog jumps on the number line: He starts at 
, then jumps in some order on each of the numbers 
exactly once and finally returns with his last jump to . Can the total distance he travelled with these 
jumps be a) 
, b) 
?
, then jumps in some order on each of the numbers 
exactly once and finally returns with his last jump to . Can the total distance he travelled with these 
jumps be a) 
, b) 
?
2 replies
Geometry
Captainscrubz 2
N
by MrdiuryPeter
Source: Own
Let 
be any point on side of .Let 
and 
be points on 
and such that 
and 
respectively. Prove that the locus of circumcenter of 
is a line.
Prove without using moving points :D
be any point on side of .Let 
and 
be points on 
and such that 
and 
respectively. Prove that the locus of circumcenter of 
is a line.Prove without using moving points :D
2 replies
inequality ( 4 var
SunnyEvan 4
N
by SunnyEvan
Let 
, such that 
Prove that :
equality cases : ?
, such that 
Prove that :
equality cases : ?
4 replies
such that ![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
2025 - Turkmenistan National Math Olympiad
A_E_R 4
N
by NODIRKHON_UZ
Source: Turkmenistan Math Olympiad - 2025
Let k,m,n>=2 positive integers and GCD(m,n)=1, Prove that the equation has infinitely many solutions in distict positive integers: x_1^m+x_2^m+⋯x_k^m=x_(k+1)^n
4 replies
9x9 board
oneplusone 8
N
by lightsynth123
Source: Singapore MO 2011 open round 2 Q2
If 46 squares are colored red in a 
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
8 replies
