are positive isn't necessary.
Y by PikaPika999, RainbowJessa
Prove that 
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Topic
First Poster
Last Poster
n-variable inequality
bakkune 0
Source: Own
Prove that the following inequality holds for all positive integer 
and all real numbers 
:
and all real numbers 
:
0 replies
5-th powers is a no-go - JBMO Shortlist
WakeUp 9
N
by Namisgood
Prove that there are are no positive integers 
and 
such that 
.
and 
such that 
.9 replies
IMO 2008, Question 1
orl 156
N
by Siddharthmaybe
Source: IMO Shortlist 2008, G1
Let 
be the orthocenter of an acute-angled triangle 
. The circle 
centered at the midpoint of 
and passing through intersects the sideline at points 
and 
. Similarly, define the points 
, 
, 
and 
.
Prove that the six points, , , , and are concyclic.
Author: Andrey Gavrilyuk, Russia
be the orthocenter of an acute-angled triangle 
. The circle 
centered at the midpoint of 
and passing through intersects the sideline at points 
and 
. Similarly, define the points 
, 
, 
and 
.Prove that the six points
, , , , and are concyclic.Author: Andrey Gavrilyuk, Russia
156 replies
APMO 2015 P1
aditya21 63
N
by Siddharthmaybe
Source: APMO 2015
Let 
be a triangle, and let 
be a point on side 
. A line through intersects side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point 
. The lines 
and 
intersect again at 
and 
respectively.
Prove that
Proposed by Warut Suksompong, Thailand
be a triangle, and let 
be a point on side 
. A line through intersects side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point 
. The lines 
and 
intersect again at 
and 
respectively.Prove that
Proposed by Warut Suksompong, Thailand
63 replies
AD=BE implies ABC right
v_Enhance 118
N
by Siddharthmaybe
Source: European Girl's MO 2013, Problem 1
The side of the triangle is extended beyond 
to so that 
. The side 
is extended beyond 
to 
so that 
. Prove that, if 
, then the triangle is right-angled.
of the triangle is extended beyond 
to so that 
. The side 
is extended beyond 
to 
so that 
. Prove that, if 
, then the triangle is right-angled.
118 replies
Cyclic Quads and Parallel Lines
gracemoon124 17
N
by Siddharthmaybe
Source: 2015 British Mathematical Olympiad?
Let 
be a cyclic quadrilateral. Let 
be the midpoint of the arc of its circumcircle which does not contain or . Let the lines 
and meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
be a cyclic quadrilateral. Let 
be the midpoint of the arc of its circumcircle which does not contain or . Let the lines 
and meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
17 replies
Problem 1 (First Day)
Valentin Vornicu 137
N
by Siddharthmaybe
1. Let be an acute-angled triangle with 
. The circle with diameter intersects the sides and at 
and 
respectively. Denote by 
the midpoint of the side . The bisectors of the angles 
and 
intersect at 
. Prove that the circumcircles of the triangles 
and 
have a common point lying on the side .
be an acute-angled triangle with 
. The circle with diameter intersects the sides and at 
and 
respectively. Denote by 
the midpoint of the side . The bisectors of the angles 
and 
intersect at 
. Prove that the circumcircles of the triangles 
and 
have a common point lying on the side .
137 replies
Concentric Circles
MithsApprentice 62
N
by Siddharthmaybe
Source: USAMO 1998
Let 
and 
be concentric circles, with in the interior of . From a point on one draws the tangent to (
). Let be the second point of intersection of and , and let be the midpoint of . A line passing through intersects at and in such a way that the perpendicular bisectors of 
and intersect at a point on . Find, with proof, the ratio 
.
and 
be concentric circles, with in the interior of . From a point on one draws the tangent to (
). Let be the second point of intersection of and , and let be the midpoint of . A line passing through intersects at and in such a way that the perpendicular bisectors of 
and intersect at a point on . Find, with proof, the ratio 
.
62 replies
four points lie on a circle
pohoatza 77
N
by Siddharthmaybe
Source: IMO Shortlist 2006, Geometry 2, AIMO 2007, TST 1, P2
Let 
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](//latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points , , 
and 
are concyclic.
Proposed by Vyacheslev Yasinskiy, Ukraine
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points , , 
and 
are concyclic.Proposed by Vyacheslev Yasinskiy, Ukraine
77 replies
Hard combi
EeEApO 6
N
by aidan0626
In a quiz competition, there are a total of 
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
6 replies
The familiar right angle from the orthocenter
buratinogigle 0
Source: Own, HSGSO P6
Let be a triangle inscribed in a circle with orthocenter 
and altitude 
. Let be the midpoint of 
. Line 
meets again at . Line 
meets again at . Let 
be the orthogonal projection of on the line . Line 
meets again at 
. Prove that 
.
be a triangle inscribed in a circle with orthocenter 
and altitude 
. Let be the midpoint of 
. Line 
meets again at . Line 
meets again at . Let 
be the orthogonal projection of on the line . Line 
meets again at 
. Prove that 
.
0 replies
ALGEBRA INEQUALITY
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Y by PikaPika999, RainbowJessa
.
Share some better sols.
Now expand and finish by AM-GM or by MuriheadShare some better sols.
This post has been edited 3 times. Last edited by Tony_stark0094, Apr 23, 2025, 12:53 AM
Reason: fjajja
Reason: fjajja
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to both sides to obtain ![\[\sum_{cyc}\frac{(a+b)(c+a)}{(b+c)} \ge 2(a+b+c).\]](http://latex.artofproblemsolving.com/b/1/e/b1e1674cee0e7445e5e2a8e8d3b201fd49247a65.png)
Substitute 
,
,
to turn this into ![\[\sum_{cyc} \frac{xz}{y} \ge \sum_{cyc} x,\]](http://latex.artofproblemsolving.com/2/e/e/2ee5eb58e69757f7815b75e979304b5f50d28c47.png)
which is clear by Muirhead.
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