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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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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

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0 replies
jwelsh
Jul 1, 2025
0 replies
P-Adic MathDash Problem
LilKirb   1
N 41 minutes ago by LilKirb
Let $N = 2^{23} - 36.$ Given that $2^{21} - 9$ is a prime, find the number of nonnegative integers $0 \leq x \leq N$ such that $N$ is a divisor of $\binom{N}{x}.$

Express the answer in the form $a \cdot b^c - d,$ where $a,b,c,d$ are positive integers, $a$ is not divisible by $b,$ and $b$ is as small as possible with $b\neq1$
1 reply
LilKirb
Yesterday at 12:24 PM
LilKirb
41 minutes ago
Original Question #8
Siopao_Enjoyer   3
N 3 hours ago by Kempu33334
Let $x$, $y$, $z$ be real numbers such that:
\[
    \begin{cases}
        x+y+z&=1\\
        x^2+y^2+z^2&=12 \\
        x^3+y^3+z^3&=18
    \end{cases}
    \]Find the value of $xyz$.

Answer Confirmation
3 replies
Siopao_Enjoyer
4 hours ago
Kempu33334
3 hours ago
[PMO26 Qualifying II.12] Equality
kae_3   4
N 3 hours ago by mathprodigy2011
The real numbers $x,y$ are such that $x\neq y$ and \[\frac{x}{26-x^2}=\frac{y}{26-y^2}=\frac{xy}{26-(xy)^2}.\]What is $x^2+y^2$?

$\text{(a) }626\qquad\text{(b) }650\qquad\text{(c) }677\qquad\text{(d) }729$

Answer Confirmation
4 replies
kae_3
Feb 21, 2025
mathprodigy2011
3 hours ago
a cute combinatorics (?) problem
pzzd   5
N 4 hours ago by pzzd
here’s a cute little problem that can be solved with combinatorics, but is also related to some very common sequences in mathematics :3

say you have a $2$-inch-wide rectangle of some length $l$, and a bunch of $2$x$1$ dominos. how many different ways can you completely cover the rectangle with dominos? you can place the dominos horizontally or vertically - for example, for a $2$-by-$3$ rectangle, a valid arrangement of dominos is $1$ vertical domino on the left and $2$ horizontal dominos on the right.

hope you find this interesting!
5 replies
pzzd
Yesterday at 2:48 PM
pzzd
4 hours ago
Digits problem
menseggerofgod   0
5 hours ago
Jean came up with a positive integer that is divisible by 13, whose digits are not zero and distinct two by two. He noticed that in this number two digits can be interchanged so that the result is also divisible by 13. What is the smallest number of digits that Jean's number could have?
0 replies
menseggerofgod
5 hours ago
0 replies
[PMO27 Areas] I.8 Radical equations
aops-g5-gethsemanea2   9
N 5 hours ago by Siopao_Enjoyer
Positive real numbers $x$ and $y$ satisfy $\sqrt x+\sqrt y=4$ and $\sqrt{x+2}+\sqrt{y+2}=5$. If $x+y=m/n$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$?
9 replies
aops-g5-gethsemanea2
Jan 25, 2025
Siopao_Enjoyer
5 hours ago
Trigonometry equation practice
ehz2701   6
N Yesterday at 7:56 PM by vanstraelen
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.

Leaderboard

problem set 1a

problem set 2a

problem set 2b
answers 2b

General techniques so far:

Trick 1: one thing to keep in mind is

[center] $\frac{1}{2} = \cos 36 - \sin 18$. [/center]

Many of these seem to be reducible to this. The half can be written as $\cos 60 = \sin 30$, and $\cos 36 = \sin 54$, $\sin 18 = \cos 72$. This is proven in solution 1a-1. We will refer to this as Trick 1.
6 replies
ehz2701
Saturday at 8:48 AM
vanstraelen
Yesterday at 7:56 PM
Easy geometry problem
menseggerofgod   5
N Yesterday at 6:27 PM by ehz2701
ABC is a right triangle, right at B, in which the height BD is drawn. E is a point on side BC such that AE = EC = 8. If BD is 6 and DE = k , find k
5 replies
menseggerofgod
Yesterday at 2:47 AM
ehz2701
Yesterday at 6:27 PM
Geometry easy
AlexCenteno2007   4
N Yesterday at 5:49 PM by AlexCenteno2007
In triangle ABC, if angle B=120°, AB=5u and BC=15u. Draw the interior bisector BE. Calculate BE
4 replies
AlexCenteno2007
Jul 11, 2025
AlexCenteno2007
Yesterday at 5:49 PM
Select 3 frm {1,2,..,4n}, 4 divides their sum
Sayan   12
N Yesterday at 4:14 PM by ParthivCalculus
Find the number of ways in which three numbers can be selected from the set $\{1,2,\cdots ,4n\}$, such that the sum of the three selected numbers is divisible by $4$.
12 replies
Sayan
May 9, 2012
ParthivCalculus
Yesterday at 4:14 PM
[JBMO 2013/3]
arcticfox009   2
N Yesterday at 3:39 PM by DAVROS
Show that

\[ \left( a + 2b + \frac{2}{a + 1} \right) \left( b + 2a + \frac{2}{b + 1} \right) \geq 16 \]
for all positive real numbers $a$ and $b$ such that $ab \geq 1$.
2 replies
arcticfox009
Jul 11, 2025
DAVROS
Yesterday at 3:39 PM
10 Problems
Sedro   8
N Yesterday at 3:34 PM by Sedro
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 $995$.
8 replies
Sedro
Jul 10, 2025
Sedro
Yesterday at 3:34 PM
angle chasing question
mahi.314   6
N Yesterday at 3:29 PM by sunken rock
Hi! I'm not comfortable with latex yet so bear with me please.
Q. in triABC, BD and CE are the bisectors of angles B,C cutting CA, AB at D,E respectively. if angle BDE= 24deg and angle CED= 18deg, find the angles of triABC.
I did find out angle A which comes out to be Click to reveal hidden text
but i'm stuck on the other two. help would be appreciated.
thanks!
6 replies
mahi.314
Jul 10, 2025
sunken rock
Yesterday at 3:29 PM
AM-GM Problem
arcticfox009   13
N Yesterday at 3:06 PM by nudinhtien
Let $x, y$ be positive real numbers such that $xy \geq 1$. Find the minimum value of the expression

\[ \frac{(x^2 + y)(x + y^2)}{x + y}. \]
answer confirmation
13 replies
arcticfox009
Jul 11, 2025
nudinhtien
Yesterday at 3:06 PM
Word problem of Absolute-Value Function
Sukardy   1
N May 29, 2025 by Mathzeus1024
Please help me solve the following word problems. It is about absolute-value function. Any sketch of function's graph will be greatly appreciated.

Problem:
A combat plane is flying $50$ m above the sea. The plane will aim an enemy submarine in the distance of $100$ m. To be aimed on the target, a missile shot by the plane plunges into the sea with a depth of $30$ m. Suppose the surface of the sea to be $X$-axis and the plane's motion is represented by a function $f(x) = p|x|+q$ for $p, q \in \mathbb{R}$. Find the value of $p + q$.
1 reply
Sukardy
Nov 6, 2019
Mathzeus1024
May 29, 2025
Word problem of Absolute-Value Function
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Sukardy
268 posts
#1 • 1 Y
Y by Adventure10
Please help me solve the following word problems. It is about absolute-value function. Any sketch of function's graph will be greatly appreciated.

Problem:
A combat plane is flying $50$ m above the sea. The plane will aim an enemy submarine in the distance of $100$ m. To be aimed on the target, a missile shot by the plane plunges into the sea with a depth of $30$ m. Suppose the surface of the sea to be $X$-axis and the plane's motion is represented by a function $f(x) = p|x|+q$ for $p, q \in \mathbb{R}$. Find the value of $p + q$.
This post has been edited 2 times. Last edited by Sukardy, Nov 6, 2019, 7:42 AM
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Mathzeus1024
1057 posts
#2
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Let the plane be at $P(0,50)$ and the sub at $S(100,-30)$ in the $xy-$plane. The function in question computes to $\textcolor{red}{f(x)=-\frac{4}{5}|x| + 50}$.
This post has been edited 1 time. Last edited by Mathzeus1024, May 29, 2025, 11:11 AM
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