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High School Olympiads
Regional, national, and international math olympiads
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Italian WinterCamps test07 Problem4
mattilgale 90
N
43 minutes ago
by mathwiz_1207
Source: ISL 2006, G3, VAIMO 2007/5
Let
be a convex pentagon such that
The diagonals
and
meet at
. Prove that the line
bisects the side
.
Proposed by Zuming Feng, USA

![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)





Proposed by Zuming Feng, USA
90 replies
Iran TST P8
TheBarioBario 8
N
an hour ago
by Mysteriouxxx
Source: Iranian TST 2022 problem 8
In triangle
, with
,
is the incenter,
is the intersection of
-excircle and
. Point
lies on the external angle bisector of
such that
and
lieas on the same side of the line
and
. Point
lies on
such that
. Circle
is tangent to
and
at
, circle
is tangent to
and
at
and both circles pass through the inside of triangle
. if
is the Midpoint od the arc
, which does not contain
, prove that
lies on the radical axis of
and
.
Proposed by Amirmahdi Mohseni






























Proposed by Amirmahdi Mohseni
8 replies
IMO 2010 Problem 6
mavropnevma 42
N
2 hours ago
by awesomeming327.
Let
be a sequence of positive real numbers, and
be a positive integer, such that
![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](//latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
Prove there exist positive integers
and
, such that
![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](//latex.artofproblemsolving.com/b/0/b/b0b2bd4507621302eeee27ae50ed9c0afd72c6ef.png)
Proposed by Morteza Saghafiyan, Iran


![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](http://latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
Prove there exist positive integers


![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](http://latex.artofproblemsolving.com/b/0/b/b0b2bd4507621302eeee27ae50ed9c0afd72c6ef.png)
Proposed by Morteza Saghafiyan, Iran
42 replies
1 viewing
PJ // AC iff BC^2 = AC· QC
parmenides51 1
N
3 hours ago
by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1998 OMM P5
The tangents at points
and
on a given circle meet at point
. Let
be a point on segment
and let
meet the circle again at
. The line through
parallel to
intersects
at
. Prove that
is parallel to
if and only if
.














1 reply
Self-evident inequality trick
Lukaluce 10
N
3 hours ago
by ytChen
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let
, and
be positive real numbers, such that
. Prove the inequality
When does the equality hold?



![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
10 replies
Power Of Factorials
Kassuno 181
N
3 hours ago
by SomeonecoolLovesMaths
Source: IMO 2019 Problem 4
Find all pairs
of positive integers such that
Proposed by Gabriel Chicas Reyes, El Salvador

![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
181 replies
Gergonne point Harmonic quadrilateral
niwobin 4
N
4 hours ago
by on_gale
Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
4 replies
NCG Returns!
blacksheep2003 64
N
4 hours ago
by SomeonecoolLovesMaths
Source: USEMO 2020 Problem 1
Which positive integers can be written in the form
for positive integers
,
,
?
![\[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]](http://latex.artofproblemsolving.com/1/e/8/1e8b3e2d9a4d0917ec622449b058702700542f22.png)



64 replies
Binomial stuff
Arne 2
N
4 hours ago
by Speedysolver1
Source: Belgian IMO preparation
Let
be prime, let
be a positive integer, show that


![\[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]](http://latex.artofproblemsolving.com/9/d/4/9d409ce186320241eb358b5fbbb8a5a66120899c.png)
2 replies
Geometry hard problem
noneofyou34 1
N
4 hours ago
by Lil_flip38
Let ABC be a triangle with incircle Γ. The tangency points of Γ with sides BC, CA, AB are A1, B1, C1 respectively. Line B1C1 intersects line BC at point A2. Similarly, points B2 and C2 are constructed. Prove that the perpendicular lines from A2, B2, C2 to lines AA1, BB1, CC1 respectively are concurret.
1 reply
