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Sharygin 2025 CR P18
Gengar_in_Galar 5
N
by hectorleo123
Source: Sharygin 2025
Let 
be a quadrilateral such that the excircles 
and 
of triangles 
and 
touching their sides 
and 
respectively touch the extension of 
at the same point 
. The segment 
meets at point 
, and the line meets at 
and 
. Prove that one of angles 
and 
is right
Proposed by: I.Kukharchuk
be a quadrilateral such that the excircles 
and 
of triangles 
and 
touching their sides 
and 
respectively touch the extension of 
at the same point 
. The segment 
meets at point 
, and the line meets at 
and 
. Prove that one of angles 
and 
is rightProposed by: I.Kukharchuk
5 replies
BMO 2025
GreekIdiot 10
N
by tranducphat
Does anyone have the problems? They should have finished by now.
10 replies
Infinitely many numbers of a given form
Stefan4024 19
N
by cursed_tangent1434
Source: EGMO 2016 Day 2 Problem 6
Let be the set of all positive integers 
such that 
has a divisor in the range 
. Prove that there are infinitely many elements of of each of the forms 
and no elements of of the form 
and 
, where 
is an integer.
be the set of all positive integers 
such that 
has a divisor in the range 
. Prove that there are infinitely many elements of of each of the forms 
and no elements of of the form 
and 
, where 
is an integer.
19 replies
Very easy case of a folklore polynomial equation
Assassino9931 1
N
by iamnotgentle
Source: Bulgaria EGMO TST 2025 P6
Determine all polynomials 
of odd degree with real coefficients such that 
.
of odd degree with real coefficients such that 
.
1 reply
Process on scalar products and permutations
Assassino9931 2
N
by Assassino9931
Source: RMM Shortlist 2024 C1
Fix an integer 
. Consider 
real numbers 
and 
. Let be the set of all pairs 
of real numbers for which 
, 
are pairwise distinct. For every such pair sort the corresponding values 
increasingly and let 
be the 
-th term in the list thus sorted. This denes an index permutation of 
. Let 
be the number of all such permutations, as the pairs run through all of . In terms of , determine the largest value may achieve over all possible choices of 
.
. Consider 
real numbers 
and 
. Let be the set of all pairs 
of real numbers for which 
, 
are pairwise distinct. For every such pair sort the corresponding values 
increasingly and let 
be the 
-th term in the list thus sorted. This denes an index permutation of 
. Let 
be the number of all such permutations, as the pairs run through all of . In terms of , determine the largest value may achieve over all possible choices of 
.
2 replies
Square problem
Jackson0423 2
N
by Jackson0423
Construct a square such that the distances from an interior point to the vertices (in clockwise order) are
1,7,8,4 respectively.
1,7,8,4 respectively.
2 replies
IMO Shortlist Problems
ABCD1728 2
N
by ABCD1728
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
2 replies
Estimate on number of progressions
Assassino9931 1
N
by BlizzardWizard
Source: RMM Shortlist 2024 C4
Let be a positive integer. For a set of real numbers, let 
denote the number of increasing arithmetic progressions of length at least two all of whose terms are in . Prove that, if is a set of real numbers, then
![\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]](//latex.artofproblemsolving.com/c/8/8/c887443e08b9f61e5ce55c3e35fcf2912a6da2a5.png)
be a positive integer. For a set of real numbers, let 
denote the number of increasing arithmetic progressions of length at least two all of whose terms are in . Prove that, if is a set of real numbers, then![\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]](http://latex.artofproblemsolving.com/c/8/8/c887443e08b9f61e5ce55c3e35fcf2912a6da2a5.png)
1 reply
2^x+3^x = yx^2
truongphatt2668 10
N
by MittenpunktpointX9
Prove that the following equation has infinite integer solutions:
10 replies
find the radius of circumcircle!
jennifreind 1
N
by ricarlos
In 
, 
is acute, 
, and 
. Let point 
be the intersection of the tangent to the circumcircle of at point 
and the perpendicular bisector of segment 
. Given that 
, find the radius of the circumcircle of 
.
IMAGE
, 
is acute, 
, and 
. Let point 
be the intersection of the tangent to the circumcircle of at point 
and the perpendicular bisector of segment 
. Given that 
, find the radius of the circumcircle of 
.IMAGE
1 reply
