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HK bisect QS
lssl 24
N
by LeYohan
Source: 1998 HK
In a concyclic quadrilateral 
,
, 
are perpendicular foot from 
to sides 
, prove that 
bisect segment
.
,
, 
are perpendicular foot from 
to sides 
, prove that 
bisect segment
.
24 replies
Points in general position
AshAuktober 3
N
by blackbluecar
Source: 2025 Nepal ptst p1 of 4
Shining tells Prajit a positive integer 
. Prajit then tries to place n points such that no four points are concyclic and no 
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
(Prajit Adhikari, Nepal and Shining Sun, USA)
. Prajit then tries to place n points such that no four points are concyclic and no 
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?(Prajit Adhikari, Nepal and Shining Sun, USA)
3 replies
and 
Prove that
![$$ a+b+kc^2\geq\frac{3\sqrt[3]{k}}{4}$$](http://latex.artofproblemsolving.com/5/5/b/55be0378562556d1aa4400bef11d1510ff722854.png)
Where 
IMO 2014 Problem 1
Amir Hossein 132
N
by maromex
Let 
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that
![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](//latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.
132 replies
IMO Genre Predictions
ohiorizzler1434 31
N
by ohiorizzler1434
Everybody, with IMO upcoming, what are you predictions for the problem genres?
Personally I predict:
Personally I predict:
31 replies
weird FE
tobiSALT 10
N
by NicoN9
Source: Pan American Girls' Mathematical Olympiad 2024, P5
Find all functions 
such that
for all real numbers
.
such that
for all real numbers
.
10 replies
2015 solutions for quotient function!
raxu 49
N
by blueprimes
Source: TSTST 2015 Problem 5
Let 
denote the number of positive integers less than 
that are relatively prime to . Prove that there exists a positive integer 
for which the equation 
has at least 
solutions in .
Proposed by Iurie Boreico
denote the number of positive integers less than 
that are relatively prime to . Prove that there exists a positive integer 
for which the equation 
has at least 
solutions in .Proposed by Iurie Boreico
49 replies
Cosine of polynomial is polynomial of cosine
yofro 1
N
by yofro
Source: 2025 HMIC #2
Find all polynomials 
with real coefficients for which there exists a polynomial with real coefficients such that for all real 
, 
with real coefficients for which there exists a polynomial with real coefficients such that for all real 
, 
1 reply
Problem 1
randomusername 72
N
by blueprimes
Source: IMO 2015, Problem 1
We say that a finite set 
of points in the plane is balanced if, for any two different points 
and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points in such that 
.
(a) Show that for all integers
, there exists a balanced set consisting of points.
(b) Determine all integers for which there exists a balanced centre-free set consisting of points.
Proposed by Netherlands
of points in the plane is balanced if, for any two different points 
and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points in such that 
.(a) Show that for all integers
, there exists a balanced set consisting of points.(b) Determine all integers
for which there exists a balanced centre-free set consisting of points.Proposed by Netherlands
72 replies
Permutation with no two prefix sums dividing each other
Assassino9931 2
N
by Assassino9931
Source: Bulgaria Team Contest, March 2025, oVlad
Does there exist an infinite sequence of positive integers 
, such that every positive integer appears exactly once as a member of the sequence and 
divides 
if and only if 
?
, such that every positive integer appears exactly once as a member of the sequence and 
divides 
if and only if 
?
2 replies
