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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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My first proof problem
OWOW   3
N 29 minutes ago by TedBot
In the equation $\sum_{i=0}^n i$ prove (or disprove) that there exists infinitely many positive integers n in which $\sum_{i=0}^n i$ sums to k! where k is a positive integer. (examples, 1+2+3=3! and 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15=5!

Also this is my first proof problem so don't get mad at me if it's really bad. (Technically I'm a middle schooler so I should post this in the middle school threads but I'm thinking proofs are more high school level so I'm posting it here.)
3 replies
OWOW
Yesterday at 11:44 PM
TedBot
29 minutes ago
J
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10 Problems
Sedro   56
N an hour ago by MathsII-enjoy
Title says most of it. I've been meaning to post a problem set on HSM since at least a few months ago, but since I proposed the most recent problems I made to the 2025 SSMO, I had to wait for that happen. (Hence, most of these problems will probably be familiar if you participated in that contest, though numbers and wording may be changed.) The problems are very roughly arranged by difficulty. Enjoy!

Problem 1: An sequence of positive integers $u_1, u_2, \dots, u_8$ has the property for every positive integer $n\le 8$, its $n^\text{th}$ term is greater than the mean of the first $n-1$ terms, and the sum of its first $n$ terms is a multiple of $n$. Let $S$ be the number of such sequences satisfying $u_1+u_2+\cdots + u_8 = 144$. Compute the remainder when $S$ is divided by $1000$.

Problem 2 (solved by fruitmonster97): Rhombus $PQRS$ has side length $3$. Point $X$ lies on segment $PR$ such that line $QX$ is perpendicular to line $PS$. Given that $QX=2$, the area of $PQRS$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Problem 3 (solved by Math-lover1): Positive integers $a$ and $b$ satisfy $a\mid b^2$, $b\mid a^3$, and $a^3b^2 \mid 2025^{36}$. If the number of possible ordered pairs $(a,b)$ is equal to $N$, compute the remainder when $N$ is divided by $1000$.

Problem 4 (solved by CubeAlgo15): Let $ABC$ be a triangle. Point $P$ lies on side $BC$, point $Q$ lies on side $AB$, and point $R$ lies on side $AC$ such that $PQ=BQ$, $CR=PR$, and $\angle APB<90^\circ$. Let $H$ be the foot of the altitude from $A$ to $BC$. Given that $BP=3$, $CP=5$, and $[AQPR] = \tfrac{3}{7} \cdot [ABC]$, the value of $BH\cdot CH$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Problem 5 (solved by maromex): Anna has a three-term arithmetic sequence of integers. She divides each term of her sequence by a positive integer $n>1$ and tells Bob that the three resulting remainders are $20$, $52$, and $R$, in some order. For how many values of $R$ is it possible for Bob to uniquely determine $n$?

Problem 6: There is a unique ordered triple of positive reals $(x,y,z)$ satisfying the system of equations \begin{align*} x^2 + 9 &= (y-\sqrt{192})^2 + 4 \\ y^2 + 4 &= (z-\sqrt{192})^2 + 49 \\ z^2 + 49 &= (x-\sqrt{192})^2 + 9. \end{align*}The value of $100x+10y+z$ can be expressed as $p\sqrt{q}$, where $p$ and $q$ are positive integers such that $q$ is square-free. Compute $p+q$.

Problem 7 (solved by sami1618): Let $S$ be the set of all monotonically increasing six-term sequences whose terms are all integers between $0$ and $6$ inclusive. We say a sequence $s=n_1, n_2, \dots, n_6$ in $S$ is symmetric if for every integer $1\le i \le 6$, the number of terms of $s$ that are at least $i$ is $n_{7-i}$. The probability that a randomly chosen element of $S$ is symmetric is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.

Problem 8: For a positive integer $n$, let $r(n)$ denote the value of the binary number obtained by reading the binary representation of $n$ from right to left. Find the smallest positive integer $k$ such that the equation $n+r(n)=2k$ has at least ten positive integer solutions $n$.

Problem 9 (solved by Math-lover1, sami1618): Let $p$ be a quadratic polynomial with a positive leading coefficient. There exists a positive real number $r$ such that $r < 1 < \tfrac{5}{2r} < 5$ and $p(p(x)) = x$ for $x \in \{ r,1, \tfrac{5}{2r} , 5\}$. Compute $p(20)$.

Problem 10 (solved by aaravdodhia, sami1618): Find the number of ordered triples of positive integers $(a,b,c)$ such that $a+b+c=995$ and $ab+bc+ca$ is a multiple of $995$.
56 replies
Sedro
Jul 10, 2025
MathsII-enjoy
an hour ago
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Inequalitis
sqing   11
N 3 hours ago by sqing
Let $ a,b,c\geq 0 , a^2+b^2+c^2 =3.$ Prove that
$$a^3 +b^3 +c^3 +\frac{11}{5}abc \leq \frac{26}{5}$$
11 replies
sqing
May 31, 2025
sqing
3 hours ago
J
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Inequalities of integers
nhathhuyyp5c   3
N 4 hours ago by Pal702004
Let $m,n$ be positive integers, $m$ is even such that $\sqrt{2}<\dfrac{m}{n}<\sqrt{2}+\dfrac{1}{2}$. Prove that there exist positive integers $k,l$ satisfying $$\left|\frac{k}{l}-\sqrt{2}\right|<\frac{m}{n}-\sqrt{2}.$$
3 replies
nhathhuyyp5c
Jun 14, 2025
Pal702004
4 hours ago
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No more topics!
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Compilation of functions problems
Saucepan_man02   5
N May 12, 2025 by Konigsberg
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
5 replies
Saucepan_man02
May 7, 2025
Konigsberg
May 12, 2025
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Compilation of functions problems
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Reveal topic tags
function
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Saucepan_man02
1408 posts
Saucepan_man02
#1 h  May 7, 2025, 6:15 AM
Y by
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
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derekli
57 posts
derekli
#2 h  May 7, 2025, 8:14 PM
Y by
Saucepan_man02 wrote:
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..

Once the stellar learning tagging system is complete (stellarlearning.app/competitive), we will have the ability to practice any problems and filter out by topics.
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Saucepan_man02
1408 posts
Saucepan_man02
#3 h  May 10, 2025, 12:45 AM
Y by
@above Kindly update here when its done :coolspeak: .

Could anyone kindly post some handout/compilation of problems for functions meanwhile?
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Saucepan_man02
1408 posts
Saucepan_man02
#4 h  May 12, 2025, 3:29 AM
Y by
\bump 8char
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lightsbug
8 posts
lightsbug
#5 h  May 12, 2025, 6:39 AM
Y by
What kind of function-related problems are you looking for? You might find something here.
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Konigsberg
2240 posts
Konigsberg
#6 h  May 12, 2025, 7:16 PM
Y by
Any resource under “Algebra” here should have a chapter on functions: https://tinyurl.com/ContestGuideIntlGDrive

One resource in particular linked there, the “AoPS MegaStore”, has a lot of handouts sorted by subject.
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