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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Max and Min
Butterfly   0
39 minutes ago

Let $a_1,a_2,\cdots,a_n$ be an arrangement of $\{1,2,3,\cdots,n\}$. Find the maximum and minimum values of $$\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}.$$
0 replies
+1 w
Butterfly
39 minutes ago
0 replies
Cup of Combinatorics
M11100111001Y1R   4
N 43 minutes ago by sami1618
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) 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.

$a)$ 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 \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
4 replies
M11100111001Y1R
May 27, 2025
sami1618
43 minutes ago
(c^n+1)/(2^na+b) is an integer for all n
parmenides51   2
N an hour ago by Assassino9931
Source: Ukraine TST 2010 p6
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.
2 replies
parmenides51
May 4, 2020
Assassino9931
an hour ago
Nine point circle + Perpendicularities
YaoAOPS   18
N an hour ago by AndreiVila
Source: 2025 CTST P2
Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$. Let $P$ be an arbitrary point on the nine point circle of $ABC$. The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$. The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$. If $M$ is the midpoint of $AQ$, show that $HX \perp DM$.
18 replies
YaoAOPS
Mar 5, 2025
AndreiVila
an hour ago
Inequality conjecture
RainbowNeos   0
an hour ago
Show (or deny) that there exists an absolute constant $C>0$ that, for all $n$ and $n$ positive real numbers $x_i ,1\leq i \leq n$, there is
\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]
0 replies
RainbowNeos
an hour ago
0 replies
inequality 2905
pennypc123456789   0
an hour ago
Consider positive real numbers \( x, y, z \) that satisfy the condition
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3.
\]Find the maximum value of the expression
\[
P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}}
+ \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}}
+ \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}.
\]
0 replies
pennypc123456789
an hour ago
0 replies
Inspired by m4thbl3nd3r
sqing   3
N 2 hours ago by sqing
Source: Own
Let $  a, b,c>0,b+c>a$. Prove that$$\sqrt{\frac{a}{b+c-a}}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)}\geq 1$$$$\frac{a}{b+c-a}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)} \geq  \frac{4\sqrt 2}{3}-1$$
3 replies
sqing
Today at 3:43 AM
sqing
2 hours ago
Inspired by qrxz17
sqing   7
N 2 hours ago by sqing
Source: Own
Let $a, b,c>0 ,(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) = 27 $. Prove that $$a+b+c\geq 3\sqrt {3}$$
7 replies
sqing
6 hours ago
sqing
2 hours ago
Geometry problem
Whatisthepurposeoflife   2
N 2 hours ago 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
2 replies
Whatisthepurposeoflife
Yesterday at 1:45 PM
Whatisthepurposeoflife
2 hours ago
A Sequence of +1's and -1's
ike.chen   36
N 2 hours ago by maromex
Source: ISL 2022/C1
A $\pm 1$-sequence is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
36 replies
ike.chen
Jul 9, 2023
maromex
2 hours ago
Sequences of real numbers
brian22   92
N 3 hours ago 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
brian22
Apr 28, 2015
NicoN9
3 hours ago
Close to JMO, but not close enough
isache   11
N Today at 6:13 AM 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
isache
Yesterday at 11:37 PM
LearnMath_105
Today at 6:13 AM
AMC and AIME help
jack_ma   9
N Apr 16, 2025 by bebebe
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?
9 replies
jack_ma
Apr 16, 2025
bebebe
Apr 16, 2025
AMC and AIME help
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jack_ma
8 posts
#1
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?
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RollingPanda4616
269 posts
#2 • 1 Y
Y by jack_ma
Do alcumus and aops books, and mock past problem sets can get you a good start. :)
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jack_ma
8 posts
#3
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How do I start learning about proves? Is there a class I can take besides woot?
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RollingPanda4616
269 posts
#4 • 1 Y
Y by jack_ma
Proofs are not needed on the AMC or AIME. :)
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jack_ma
8 posts
#5
Y by
I'm aware.
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Jaxman8
124 posts
#6 • 1 Y
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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programjames1
3046 posts
#7 • 1 Y
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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RollingPanda4616
269 posts
#8 • 2 Y
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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akliu
1801 posts
#9 • 2 Y
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.
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bebebe
993 posts
#10 • 1 Y
Y by jack_ma
If you are having trouble with the concepts/big ideas, take classes on topics u haven't leanred on aops. Otherwise, it's a speed/problem solving issue, and then it's just about grinding.
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