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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
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It's not like I share elements with you or anything, baka!
fidgetboss_4000   49
N 23 minutes ago by xHypotenuse
Source: AIME I #12
For any finite set $X$, let $|X|$ denote the number of elements in $X.$ Define $$S_n = \sum |A \cap B|,$$where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, …, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$(A, B) \in \{ (\emptyset, \emptyset), (\{1\}, \{1\}), (\{1\}, \{2\}), (\{2\}, \{1\}), (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\})\},$$giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$
49 replies
fidgetboss_4000
Feb 9, 2022
xHypotenuse
23 minutes ago
J
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JSMC texas
BossLu99   65
N an hour ago by Hanruz
who is going to JSMC texas
65 replies
BossLu99
Apr 28, 2025
Hanruz
an hour ago
J
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How many positive factors does 23,232 have?
popcorn1   18
N an hour ago by Ryanzzz
Source: 2018 AMC 8 #18
How many positive factors does $23{,}232$ have?

$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }42$
18 replies
popcorn1
Nov 20, 2018
Ryanzzz
an hour ago
J
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star on a quilt
derekwang2048   23
N an hour ago by Ryanzzz
Source: 2025 AMC 8 #1
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4$-by-$4$ grid is covered by the star?

$\textbf{(A)}\ 40\qquad \textbf{(B)}\ 50\qquad \textbf{(C)}\ 60\qquad \textbf{(D)}\ 75\qquad \textbf{(E)}\ 80$
IMAGE

Thank you @zhenghua for the diagram!
23 replies
derekwang2048
Jan 30, 2025
Ryanzzz
an hour ago
J
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Estimating the Density
zqy648   1
N Today at 8:32 AM by watery
Source: 2024 May 谜之竞赛-6
Given non-empty subset \( I \) of the set of positive integers, a positive integer \( n \) is called good if for every prime factor \( p \) of \( n \), \( \nu_p(n) \in I \). For a positive real number \( x \), let \( S(x) \) denote the number of good numbers not exceeding \( x \).

Determine all positive real numbers \( C \) and \( \alpha \) such that \( \lim\limits_{x \to +\infty} \dfrac{S(x)}{x^\alpha} = C. \)

Proposed by Zhenqian Peng, High School Affiliated to Renmin University of China
1 reply
zqy648
Jul 13, 2025
watery
Today at 8:32 AM
J
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Great Use of Weyl Distribution
zqy648   1
N Today at 7:26 AM by EthanWYX2009
Source: 2025 March 谜之竞赛-6
Prove that for any real number \(\varepsilon > 0\), there exists a positive integer \(N\) such that for any prime \(p > N\) and any primitive root \(g\) modulo \(p\), if we define the set
\[ S = \left\{(i, j) \mid 1 \leq i < j \leq p - 1, \left\{ \frac{g^i}{p} \right\} > \left\{ \frac{g^j}{p} \right\} \right\} \]where \(\{x\}\) denotes the fractional part of the real number \(x\), then
\[ \left( \frac{1}{4} - \varepsilon \right) p^2 < |S| < \left( \frac{1}{4} + \varepsilon \right) p^2. \]Created by Zhenyu Dong and Chunji Wang
1 reply
zqy648
Jul 13, 2025
EthanWYX2009
Today at 7:26 AM
J
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NT By Probabilistic Method
EthanWYX2009   3
N Today at 7:20 AM by EthanWYX2009
Source: 2024 March 谜之竞赛-6
Given a positive integer \( k \) and a positive real number \( \varepsilon \), prove that there exist infinitely many positive integers \( n \) for which we can find pairwise coprime integers \( n_1, n_2, \cdots, n_k \) less than \( n \) satisfying
\[\gcd(\varphi(n_1), \varphi(n_2), \cdots, \varphi(n_k)) \geq n^{1-\varepsilon}.\]Proposed by Cheng Jiang from Tsinghua University
3 replies
EthanWYX2009
Jul 14, 2025
EthanWYX2009
Today at 7:20 AM
J
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infinite series
Martin.s   1
N Today at 4:38 AM by Svyatoslav
Can the infinite series

$$\sum\limits_{n=1}^{\infty}\frac{x^n}{n!}\ln(n+\alpha)$$
be expressed in terms of known functions and constants?
1 reply
Martin.s
Jul 13, 2025
Svyatoslav
Today at 4:38 AM
J
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Sequence
centslordm   10
N Today at 3:43 AM by megarnie
Determine the rest \[1,\,9,\,2,\,1,\,8,\,3,\,2,\,7,\,4,\,\ldots \]
10 replies
centslordm
Jun 17, 2025
megarnie
Today at 3:43 AM
J
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Double integral with sqrtx
mudkip42   1
N Yesterday at 9:33 PM by mudkip42
Source: MIT 18.02
Evaluate
\[ \int_0^1 \int_0^{\sqrt{x}} \frac{2xy}{1-y^4} \ dy \ dx. \]
1 reply
mudkip42
Yesterday at 9:32 PM
mudkip42
Yesterday at 9:33 PM
J
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an extremely difficult problem
szxz   1
N Yesterday at 8:30 PM by szxz
One class needs to choose n students to join a competition. The teacher hold n exams in order to choose n students. In each exam, there are no two students who have the same score. The teacher will hold one exam and pick the student whose score is the highest in that exam. Then, the teacher will hold another exam and pick the student whose score is the highest in this exam from the remaining students. In this way, the teacher will choose n students. Obviously, the order that the teacher hold the exams will affect the students will be chosen. So, how many students have the chance to be chosen at most?
For example, when n = 2 and in exam A, Alice is the first, Bob is the second, Tom is the third. In
exam B, Alice is the first, Tom is the second, Bob is the third. If the teacher choose exam A at first and then choose exam B, Alice and Tom will be chosen. If the teacher choose exam B at first and then choose exam A, Alice and Bob will be chosen. So Alice, Bob and Tom have the chance to be chosen.
1 reply
szxz
Yesterday at 8:27 PM
szxz
Yesterday at 8:30 PM
J
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Korea csat problem, so-called “Killer problem”..
darrime627   1
N Yesterday at 4:52 PM by Zennggg
Find the values of \( a \) and \( b \) such that the function \( f(x) \), which has a second derivative for all \( x \in \mathbb{R} \), satisfies the following conditions:

\[ (\gamma) \quad [f(x)]^5 + [f(x)]^3 + ax + b = \ln \left( x^2 + x + \frac{5}{2} \right) \]
\[ (\delta) \quad f(-3) f(3) < 0, \quad f'(2) > 0 \]
1 reply
darrime627
Jul 14, 2025
Zennggg
Yesterday at 4:52 PM
J
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Subset Ordered Pairs of {1, 2, ..., 10}
ahaanomegas   12
N Yesterday at 3:08 AM by mudkip42
Source: Putnam 1990 A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
12 replies
ahaanomegas
Jul 12, 2013
mudkip42
Yesterday at 3:08 AM
J
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Partial sum of Taylor series of e^x
NamelyOrange   2
N Yesterday at 2:09 AM by genius_007
Source: 2022 National Taiwan University STEM Development Program Admissions Test, P1
For positive integer $n$ and positive real $x$, define $e_n(x) = \sum_{k=0}^{n}\frac{x^k}{k!}$.

(a) Prove that $\frac{1}{n!}\le \frac{m^{m-n-1}}{(m-1)!}$ for all integer $n\ge m >1$.

(b) Use (a) to prove that $e_m(x)\le e_n(x)\le e_{m-1}(x)+\frac{x^m}{(m-1)!(m-x)}$ for all integer $n\ge m >1$ and real $0<x<m$.

(c) Use (b) to prove that for all positive real $x$, there exists some positive $L$ such that $e_n(x)<L$ for all positive integer $n$.

(d) Prove that for positive integer $m<n$ and positive real $x,y$ such that $x+y<m$, we have $0\le e_n(x+y)-e_n(x)e_n(y)\le \frac{m^{m-n-1}(x+y)^{n+1}}{(m-1)!(m-x-y)}$.
2 replies
NamelyOrange
Jul 15, 2025
genius_007
Yesterday at 2:09 AM
J
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J
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USAMO Prep
DPopov   8
N Apr 1, 2005 by joml88
Can anyone provide insight on how to practice for the USAMO for this year AND the future?
8 replies
DPopov
Mar 31, 2005
joml88
Apr 1, 2005
J
USAMO Prep
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DPopov
1398 posts
DPopov
#1 h  Mar 31, 2005, 3:03 AM • 2 Y
Y by Adventure10, Mango247
Can anyone provide insight on how to practice for the USAMO for this year AND the future?
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probability1.01
2743 posts
probability1.01
#2 h  Mar 31, 2005, 3:13 AM • 2 Y
Y by Adventure10, Mango247
I look at past USAMOs at http://www.unl.edu/amc . :D
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DPopov
1398 posts
DPopov
#3 h  Mar 31, 2005, 4:14 AM • 2 Y
Y by Adventure10, Mango247
I do not think that helps very much since similar problems RARELY show up again. However, I was wondering what theorems and ideas are important to know (of course one must also have a creative mind).
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darktreb
732 posts
darktreb
#4 h  Mar 31, 2005, 5:03 AM • 2 Y
Y by Adventure10, Mango247
Quote:
I do not think that helps very much since similar problems RARELY show up again.

Pretty much every inequality that uses a certain theorem or technique somewhere in the proof is similar.

Also look over the last 10 years and you'll see several problems that ask you to either create a set S that satisfies certain properties (98 #5) or prove that a set consists of all integers (04 #2, 01 #5). While different techniques are used in each case, truth is the general feel of a situation makes having the practice very useful.

Along similar lines, just about every functional equation uses the same certain techniques.
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DPopov
1398 posts
DPopov
#5 h  Mar 31, 2005, 11:13 PM • 2 Y
Y by Adventure10, Mango247
Ah, thanks! I just wanted to review somethings before I actually have to take it. Although 20 days may not seem like a lot of time, miracles can happen, and also studying wouldn't hurt :P :lol:
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darktreb
732 posts
darktreb
#6 h  Mar 31, 2005, 11:23 PM • 2 Y
Y by Adventure10, Mango247
Quote:
miracles can happen, and also studying wouldn't hurt Razz Very Happy

Only takes two problems to get to MOP usually. Even a semi-miracle would work eh?
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henryyangrui
175 posts
henryyangrui
#7 h  Mar 31, 2005, 11:51 PM • 2 Y
Y by Adventure10, Mango247
I work on AoPS book, especially on the chapters about proof, since i don't know anything about proof.
BTW, Whats the most frequent topic that appears in USAMO? Geometry or algebra?
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henryyangrui
175 posts
henryyangrui
#8 h  Apr 1, 2005, 12:01 AM • 2 Y
Y by Adventure10, Mango247
How many points is in USAMO and what's the qualifying score for IMO?
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joml88
6343 posts
joml88
#9 h  Apr 1, 2005, 12:05 AM • 3 Y
Y by Adventure10, Mango247, and 1 other user
42 points possible for the USAMO. To get on the US IMO team there is a Team Selection Test (TST) at MOP for that. The score depends on how everyone does on that. But basically you have to be top 6 to compete at the IMO and #7 is the alternate.
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