G
Topic
First Poster
Last Poster
Italian WinterCamps test07 Problem4
mattilgale 90
N
by mathwiz_1207
Source: ISL 2006, G3, VAIMO 2007/5
Let 
be a convex pentagon such that
![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](//latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
. Prove that the line 
bisects the side 
.
Proposed by Zuming Feng, USA
be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
. Prove that the line 
bisects the side 
.Proposed by Zuming Feng, USA
90 replies
Iran TST P8
TheBarioBario 8
N
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
, 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
8 replies
IMO 2010 Problem 6
mavropnevma 42
N
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
be a sequence of positive real numbers, and 
be a positive integer, such that![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](http://latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
Prove there exist positive integers
and 
, such that![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](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
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 
.
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
by ytChen
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and 
be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
10 replies
Power Of Factorials
Kassuno 181
N
by SomeonecoolLovesMaths
Source: IMO 2019 Problem 4
Find all pairs 
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](//latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
181 replies
Gergonne point Harmonic quadrilateral
niwobin 4
N
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
by SomeonecoolLovesMaths
Source: USEMO 2020 Problem 1
Which positive integers can be written in the form ![\[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]](//latex.artofproblemsolving.com/1/e/8/1e8b3e2d9a4d0917ec622449b058702700542f22.png)
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)
for positive integers 
, 
, ?
64 replies
Binomial stuff
Arne 2
N
by Speedysolver1
Source: Belgian IMO preparation
Let 
be prime, let 
be a positive integer, show that ![\[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]](//latex.artofproblemsolving.com/9/d/4/9d409ce186320241eb358b5fbbb8a5a66120899c.png)
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
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
