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Regional, national, and international math olympiads
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Topic
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Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman 65
N
37 minutes ago
by youochange
Source: BMO 2023 Problem 1
Find all functions
such that for all
,
![\[xf(x+f(y))=(y-x)f(f(x)).\]](//latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
Proposed by Nikola Velov, Macedonia


![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
Proposed by Nikola Velov, Macedonia
65 replies

3 var inequality
ehuseyinyigit 1
N
44 minutes ago
by ehuseyinyigit
Source: Own
Let
be positive real numbers. Prove that


1 reply
trig manipulation with sum of sines identity
ACalculationError 0
an hour ago
Source: PMO 2018 Qualifying Stage Part I. 10
Problem Statement: In triangle
, suppose
What is the measure of angle
?
Answer Confirmation
Solution



Answer Confirmation

Solution
Square and add the two equations:
Expanding gives
Regroup terms:
Since
,
, and
this becomes
so
Thus
or
To prove
doesn't work, if
then
so
and
which implies
contradicting the first equation. Hence,
and
.


















0 replies

Periodic sequence
EeEeRUT 9
N
an hour ago
by Supertinito
Source: Isl 2024 A5
Find all periodic sequence
of real numbers such that the following conditions hold for all
:
Proposed by Dorlir Ahmeti, Kosovo



Proposed by Dorlir Ahmeti, Kosovo
9 replies
trig basic identities
ACalculationError 0
an hour ago
Source: Sipnayan 2017 Junior High School Average #1
Problem Statement: Let
satisfy
and
. Find 
Answer Confirmation
Solution
![$x,y\in[0,\frac{\pi}{2}]$](http://latex.artofproblemsolving.com/1/6/8/1682b58d11bfb7d88fb826a857bb2bf8a8ac9969.png)



Answer Confirmation

Solution
Since
are acute, we have:
and
So:
and
Hence:


![\[\cos x = \sqrt{1-\sin^2 x} = \sqrt{1-(5/13)^2} = 12/13\]](http://latex.artofproblemsolving.com/c/9/f/c9fa5dfcd05736b8d89e40f2fa30bdba8b9bf898.png)
![\[\cos y =\sqrt{1-\sin^2 y}= \sqrt{1-(5/17)^2}= 8/17\]](http://latex.artofproblemsolving.com/8/c/0/8c01f9c21b6126105a491931a68c7e1b12df1276.png)



0 replies

I am [not] a parallelogram
peppapig_ 18
N
an hour ago
by endless_abyss
Source: ISL 2024/G4
Let
be a quadrilateral with
parallel to
and
. Lines
and
intersect at a point
. Point
distinct from
lies on the circumcircle of triangle
such that
. Point
distinct from
lies on the circumcircle of triangle
such that
. Lines
and
intersect at
.
Prove that
is parallel to
.
Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine


















Prove that


Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
18 replies
Problem 3 IMO 2005 (Day 1)
Valentin Vornicu 125
N
an hour ago
by blueprimes
Let
be three positive reals such that
. Prove that
![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](//latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
Hojoo Lee, Korea


![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
Hojoo Lee, Korea
125 replies
An easy symmetric inequality
seoneo 14
N
an hour ago
by blueprimes
Source: kjmo 2012 pr 1
Prove the following inequality where positive reals
,
,
satisfies
.




![\[
\frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
\]](http://latex.artofproblemsolving.com/4/0/f/40f8383eb6f59646ef6c41b662e9b35831664d5a.png)
14 replies
A functional equation
joybangla 14
N
an hour ago
by Lyte188
Source: Switzerland Math Olympiad, Final round 2014, P3
Find all such functions
such that for all
the following holds :


![\[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]](http://latex.artofproblemsolving.com/a/9/9/a99ec11be5ab98c79efcf7a420e46e0584c8dc01.png)
14 replies
Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance 70
N
an hour ago
by hectorleo123
Source: Taiwan 2014 TST1, Problem 3
Let
be a triangle with incenter
, and suppose the incircle is tangent to
and
at
and
. Denote by
and
the reflections of
and
over
. Let
be the intersection of
with
, and let
be the midpoint of
. Prove that
and
are perpendicular.


















70 replies
