.
.
Y by
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?
G
Topic
First Poster
Last Poster
Max and Min
Butterfly 0
Let
be an arrangement of 
. Find the maximum and minimum values of 
0 replies
+1 wCup of Combinatorics
M11100111001Y1R 4
N
by sami1618
Source: Iran TST 2025 Test 4 Problem 2
There are 
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
4 replies
(c^n+1)/(2^na+b) is an integer for all n
parmenides51 2
N
by Assassino9931
Source: Ukraine TST 2010 p6
Find all pairs of odd integers 
and 
for which there exists a natural number
such that the number 
is integer for all natural 
.
and 
for which there exists a natural number
such that the number 
is integer for all natural 
.
2 replies
Nine point circle + Perpendicularities
YaoAOPS 18
N
by AndreiVila
Source: 2025 CTST P2
Suppose 
has 
as the midpoint of 
and orthocenter 
. Let 
be an arbitrary point on the nine point circle of 
. The line through perpendicular to 
intersects at 
. The line through 
perpendicular to 
intersects 
at 
. If 
is the midpoint of , show that 
.
has 
as the midpoint of 
and orthocenter 
. Let 
be an arbitrary point on the nine point circle of 
. The line through perpendicular to 
intersects at 
. The line through 
perpendicular to 
intersects 
at 
. If 
is the midpoint of , show that 
.
18 replies
Inequality conjecture
RainbowNeos 0
Show (or deny) that there exists an absolute constant 
that, for all and positive real numbers 
, there is
![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](//latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
that, for all and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
0 replies
inequality 2905
pennypc123456789 0
Consider positive real numbers 
that satisfy the condition
![\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3.
\]](//latex.artofproblemsolving.com/2/7/d/27d266570e3a105a4b416bb7c55904ffabf7a07c.png)
Find the maximum value of the expression
![\[
P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}}
+ \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}}
+ \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}.
\]](//latex.artofproblemsolving.com/b/a/f/bafc95e59d2f28dbb4e02771db38ce736e3fd45c.png)
that satisfy the condition![\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3.
\]](http://latex.artofproblemsolving.com/2/7/d/27d266570e3a105a4b416bb7c55904ffabf7a07c.png)
Find the maximum value of the expression![\[
P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}}
+ \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}}
+ \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}.
\]](http://latex.artofproblemsolving.com/b/a/f/bafc95e59d2f28dbb4e02771db38ce736e3fd45c.png)
0 replies
Geometry problem
Whatisthepurposeoflife 2
N
by Whatisthepurposeoflife
Source: Derived from MEMO 2024 I3
Triangle ∆ABC is scalene the circle w that goes through the points A and B intersects AC at E BC at D let the Lines BE and AD intersect at point F. And let the tangents A and B of circle w Intersect at point G.
Prove that C F and G are collinear
Prove that C F and G are collinear
2 replies
A Sequence of +1's and -1's
ike.chen 36
N
by maromex
Source: ISL 2022/C1
A 
-sequence is a sequence of 
numbers 
each equal to either 
or 
. Determine the largest 
so that, for any -sequence, there exists an integer 
and indices 
so that 
for all 
, and 
-sequence is a sequence of 
numbers 
each equal to either 
or 
. Determine the largest 
so that, for any -sequence, there exists an integer 
and indices 
so that 
for all 
, and 
36 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
Close to JMO, but not close enough
isache 11
N
by LearnMath_105
Im currently a freshman in hs, and i rlly wanna make jmo in sophmore yr. Ive been cooking at in-person competitions recently (ucsd hmc, scmc, smt, mathcounts) but I keep fumbling jmo. this yr i had a 133.5 on 10b and a 9 on aime. How do i get that up by 20 points to a 240?
11 replies
AMC and AIME help
G
H
G
H
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by jack_ma
Do alcumus and aops books, and mock past problem sets can get you a good start. 
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by jack_ma
Proofs are not needed on the AMC or AIME.
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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?
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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.
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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.
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
N Quick Reply
G
H
