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Topic
First Poster
Last Poster
Zsigmondy's theorem
V0305 22
N
by CatCatHead
Is Zsigmondy's theorem allowed on the IMO, and is it allowed on the AMC series of proof competitions (e.g. USAJMO, USA TSTST)?
22 replies
happy configs
KevinYang2.71 60
N
by ray66
Source: USAJMO 2024/2
Let 
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Proposed by Serena An and Claire Zhang
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.Proposed by Serena An and Claire Zhang
60 replies
USAJMO #5 - points on a circle
hrithikguy 223
N
by MathRook7817
Points 
lie on a circle 
and point 
lies outside the circle. The given points are such that (i) lines 
and 
are tangent to , (ii) 
are collinear, and (iii) 
. Prove that 
bisects 
.
lie on a circle 
and point 
lies outside the circle. The given points are such that (i) lines 
and 
are tangent to , (ii) 
are collinear, and (iii) 
. Prove that 
bisects 
.
223 replies
Invert Your Expectations
AwesomeYRY 44
N
by akliu
Source: USAMO 2022/3
Let 
be the set of all positive real numbers. Find all functions 
such that for all 
we have
![\[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]](//latex.artofproblemsolving.com/7/9/e/79e9b17e39cd2c0466afbdc665b6d09a07d9a50e.png)
be the set of all positive real numbers. Find all functions 
such that for all 
we have![\[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]](http://latex.artofproblemsolving.com/7/9/e/79e9b17e39cd2c0466afbdc665b6d09a07d9a50e.png)
44 replies
GCD Set Condition
P_Groudon 100
N
by maromex
Source: 2021 AMO #4 / JMO #5
A finite set of positive integers has the property that, for each 
and each positive integer divisor 
of 
, there exists a unique element 
satisfying 
. (The elements and 
could be equal.)
Given this information, find all possible values for the number of elements of.
of positive integers has the property that, for each 
and each positive integer divisor 
of 
, there exists a unique element 
satisfying 
. (The elements and 
could be equal.)Given this information, find all possible values for the number of elements of
.
100 replies
Another Cubic Curve!
v_Enhance 165
N
by maromex
Source: USAMO 2015 Problem 1, JMO Problem 2
Solve in integers the equation
![\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]](//latex.artofproblemsolving.com/5/6/b/56b8cb8c13616501dd8308989cea2b338e91598b.png)
![\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]](http://latex.artofproblemsolving.com/5/6/b/56b8cb8c13616501dd8308989cea2b338e91598b.png)
165 replies
2n equations
P_Groudon 83
N
by Roots_Of_Moksha
Let 
be an integer. Find all positive real solutions to the following system of 
equations:
be an integer. Find all positive real solutions to the following system of 
equations:
83 replies
Sequences of real numbers
brian22 92
N
by NicoN9
Source: USAJMO 2015 Problem 1
Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after one initial move is applied to the sequence -- no matter what move -- there is always a way to continue with a finite sequence of moves so as to obtain in the end a constant sequence.
92 replies
usamOOK geometry
KevinYang2.71 108
N
by ray66
Source: USAMO 2025/4, USAJMO 2025/5
Let 
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
108 replies
Scary Binomial Coefficient Sum
EpicBird08 44
N
by ray66
Source: USAMO 2025/5
Determine, with proof, all positive integers 
such that 
is an integer for every positive integer 
such that 
is an integer for every positive integer 
44 replies
