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This year, I got a 3 on the AIME. I want to make olympiad by 2026. I made AIME by a small margin. To make olympiad, I need good scores on the AMC and AIME. What should I do to increase my AMC and AIME score?
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Past USAMO Medals
sdpandit 1
N
by CatCatHead
Does anyone know where to find lists of USAMO medalists from past years? I can find the 2025 list on their website, but they don't seem to keep lists from previous years and I can't find it anywhere else. Thanks!
1 reply
interesting functional
Pomegranat 2
N
by Pomegranat
Source: I don't know sorry
Find all functions 
such that for all positive real numbers 
and 
, the following equation holds:
![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](//latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all positive real numbers 
and 
, the following equation holds:![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
2 replies
Function equation algebra
TUAN2k8 1
N
by TUAN2k8
Source: Balkan MO 2025
Find all functions 
such that for all 
and 
,
such that for all 
and 
,
1 reply
Functional equation with a twist (it's number theory)
Davdav1232 1
N
by NO_SQUARES
Source: Israel TST 8 2025 p2
Prove that for all primes 
such that 
or 
, there exist integers
![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](//latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that
![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](//latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
such that 
or 
, there exist integers![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
1 reply
Coolabra
Titibuuu 3
N
by sqing
Let 
be distinct real numbers such that
![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](//latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
3 replies
Is the result of this is the same as cauchy?
ItzsleepyXD 0
Source: curiosity
Prove or disprove that for all continuous or monotonic function 
.The solution to 
is only 
for some 
.The solution to 
is only 
for some 
0 replies
Maximum area of a triangle
Kunihiko_Chikaya 1
N
by Mathzeus1024
Source: 2008 Keio University entrance exam/Medical
(1) For 
, consider the two circles on 
plane: 
. Find the range of 
for which these circles have two intersection points, then find the equation of the line passing through these points.
(2) Given points
on the perimeter of a circle with radius 1. Let the length of the cord 
. When two points move on the circle, find the maximum area of 
.
, consider the two circles on 
plane: 
. Find the range of 
for which these circles have two intersection points, then find the equation of the line passing through these points.(2) Given points
on the perimeter of a circle with radius 1. Let the length of the cord 
. When two points move on the circle, find the maximum area of 
.
1 reply
algebraic inequality
produit 1
N
by sqing
Positive a, b, c satisfy a + b + c = ab + bc + ca. Prove that
a + b + c + 1 ⩾ 4abc.
a + b + c + 1 ⩾ 4abc.
1 reply
2001-th sequence term less than 1001
orl 6
N
by Bryan0224
Source: CWMO 2001, Problem 1
The sequence 
satisfies 
. Prove that 
.
satisfies 
. Prove that 
.
6 replies
usamOOK geometry
KevinYang2.71 106
N
by jasperE3
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 
.
106 replies
AMC and AIME help
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Y by jack_ma
Do alcumus and aops books, and mock past problem sets can get you a good start. 
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Y by jack_ma
Proofs are not needed on the AMC or AIME.
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Y by jack_ma
RollingPanda4616 wrote:
Do alcumus and aops books, and mock past problem sets can get you a good start.
Alcumus is good, but it isn’t that difficult. Is there like a different alcumus with harder problems besides math dash or amc trainer?
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Y by jack_ma
jack_ma wrote:
How do I start learning about proofs? Is there a class I can take besides woot?
I think the easiest way to start proofs is to write up solutions on the AoPS forums. Solutions look very close to proofs, as long as you're not doing "fake solves" like Engineer's Induction. When you start on actual proof questions, start with easier olympiads or state competitions. For example, the Canadian Math Olympiad or the UNM-PNM. Questions on the High School Olympiads forum will usually be too difficult when you're just starting out, and besides it has a huge overrepresentation of geometry and inequalities.
This post has been edited 1 time. Last edited by programjames1, Apr 16, 2025, 1:56 AM
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Y by Jaxman8, jack_ma
Jaxman8 wrote:
RollingPanda4616 wrote:
Do alcumus and aops books, and mock past problem sets can get you a good start.
Alcumus is good, but it isn’t that difficult. Is there like a different alcumus with harder problems besides math dash or amc trainer?
I'm not aware of any alternatives. And also, Alcumus can get pretty hard. Turn the difficulty up to Insanely Hard, and do areas that you're weak in.
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Y by jack_ma, RollingPanda4616
I recommend doing OTIS. I personally started doing this after I was trying to improve my AIME score, and had zero proof-writing experience whatsoever. You'll get the hang of it pretty soon, and it'll be really fun! By doing harder problems, you also train yourself to do easier problems.
For mock scores, I'd recommend just finding some AoPSwiki mocks for the AMCs, and doing past years' mocks for the AIME.
For mock scores, I'd recommend just finding some AoPSwiki mocks for the AMCs, and doing past years' mocks for the AIME.
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