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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
J
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Ez geo with 2 circiles and parallel lines
Tofa7a._.36   1
N 5 minutes ago by aaravdodhia
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $D$ be the intersection of $\omega$ with the angle bisector of $\angle BAC$. Let $E$ be on $[DC)$ and assume that the circumcircle of the triangle $ACE$ meets $[BC]$ at $F$. Let $(DF)$ meet $\omega$ and $(ACF)$ at $G$ and $H$, respectively.
Prove that $(GC) \parallel (HE)$ .
1 reply
Tofa7a._.36
Today at 9:12 AM
aaravdodhia
5 minutes ago
J
H
Fe in Q* with nice condition
Tofa7a._.36   3
N 21 minutes ago by Tofa7a._.36
Find all surjective functions $f : \mathbb{Q^*} \to \mathbb{Q^*}$ such that: $$(f (x) + f (y))f (x + y) = f (xy)$$for all $x,y \in \mathbb{Q^*}$ with $x+y\ne 0$.
3 replies
Tofa7a._.36
Today at 9:58 AM
Tofa7a._.36
21 minutes ago
J
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10 Problems
Sedro   12
N 37 minutes ago by aaravdodhia
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 increasing sequence of positive integers $u_1, u_2, \dots, u_8$ has the property that the sum of its first $n$ terms is divisible by $n$ for every positive integer $n\le 8$. 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: 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: 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: 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: 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: 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: 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: 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 $995$.
12 replies
Sedro
Jul 10, 2025
aaravdodhia
37 minutes ago
J
H
Inversive Geo
Tofa7a._.36   0
an hour ago
Let $ABC$ be an acute and scalene triangle with circumcircle $\omega$. The perpendicular bisector of the segment $AB$ intersects the lines $BC$ and $AC$ in points $D$ and $E$, respectively, such that $E$ lies outside segment $AC$. The perpendicular to line $BC$ from $D$ intersects $(BCE)$ at a point $X$ outside $\triangle ABC$. Line $DX$ intersects line $AC$ at $Y$ and $\omega$ at points $Z$ and $T$ such that $Z$ lies on the arc $AC$ that does not contain $B$. The circumcircle of triangle $\triangle ZET$ intersects the side $BC$ and the circumcircle of triangle $\triangle YDE$ in $P$ and $Q$, respectively.
Prove that the tangent to $(YZQ)$ from $Z$, the tangent to $(YTQ)$ from $T$, and the line $PX$ meet at one point.
0 replies
Tofa7a._.36
an hour ago
0 replies
J
No more topics!
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J
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Easy Function
Darealzolt   1
N Jun 6, 2025 by alexheinis
Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
1 reply
Darealzolt
Jun 6, 2025
alexheinis
Jun 6, 2025
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Easy Function
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Darealzolt
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Darealzolt
#1 h  Jun 6, 2025, 8:47 AM
Y by
Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
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alexheinis
10741 posts
alexheinis
#2 h  Jun 6, 2025, 8:50 AM
Y by
Take $y=0$ then $f(x)=f(0)$ and $f$ is constant. Hence $f(3)=26$.
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