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Periodic sequence
EeEeRUT 9
N
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
of real numbers such that the following conditions hold for all 
:
Proposed by Dorlir Ahmeti, Kosovo
9 replies
trig basic identities
ACalculationError 0
Source: Sipnayan 2017 Junior High School Average #1
Problem Statement: Let ![$x,y\in[0,\frac{\pi}{2}]$](//latex.artofproblemsolving.com/1/6/8/1682b58d11bfb7d88fb826a857bb2bf8a8ac9969.png)
satisfy 
and 
. Find 
![$x,y\in[0,\frac{\pi}{2}]$](http://latex.artofproblemsolving.com/1/6/8/1682b58d11bfb7d88fb826a857bb2bf8a8ac9969.png)
satisfy 
and 
. Find 
are acute, we have: ![\[\cos x = \sqrt{1-\sin^2 x} = \sqrt{1-(5/13)^2} = 12/13\]](http://latex.artofproblemsolving.com/c/9/f/c9fa5dfcd05736b8d89e40f2fa30bdba8b9bf898.png)
and ![\[\cos y =\sqrt{1-\sin^2 y}= \sqrt{1-(5/17)^2}= 8/17\]](http://latex.artofproblemsolving.com/8/c/0/8c01f9c21b6126105a491931a68c7e1b12df1276.png)
So:
and 
Hence:
0 replies
+1 wI am [not] a parallelogram
peppapig_ 18
N
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
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
18 replies
Problem 3 IMO 2005 (Day 1)
Valentin Vornicu 125
N
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
be three positive reals such that 
. Prove that![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
Hojoo Lee, Korea
125 replies
An easy symmetric inequality
seoneo 14
N
by blueprimes
Source: kjmo 2012 pr 1
Prove the following inequality where positive reals 
, 
, 
satisfies 
.
![\[
\frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
\]](//latex.artofproblemsolving.com/4/0/f/40f8383eb6f59646ef6c41b662e9b35831664d5a.png)
, 
, 
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
by Lyte188
Source: Switzerland Math Olympiad, Final round 2014, P3
Find all such functions 
such that for all 
the following holds :
![\[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]](//latex.artofproblemsolving.com/a/9/9/a99ec11be5ab98c79efcf7a420e46e0584c8dc01.png)
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
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.
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
Inequality by Po-Ru Loh
v_Enhance 60
N
by blueprimes
Source: ELMO 2003 Problem 4
Let 
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](//latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](//latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
60 replies
+1 wRomania TST JBMO 2016 1
GGPiku 11
N
by lendsarctix280
Let be a acute triangle where 
. Prove that if the Euler's line of intersects 
in 
, then 
is equilateral.
be a acute triangle where 
. Prove that if the Euler's line of intersects 
in 
, then 
is equilateral.
11 replies
Inequality with abc=1
tenplusten 12
N
by blueprimes
Source: JBMO 2011 Shortlist A7
Let 
be positive reals such that 
.Prove the inequality 
12 replies
