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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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9 zeroes!.
ericheathclifffry   49
N 15 minutes ago by shaayonsamanta
Redacted
49 replies
+1 w
ericheathclifffry
May 5, 2025
shaayonsamanta
15 minutes ago
J
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Worst math problems
LXC007   94
N 20 minutes ago by cheltstudent
What is the most egregiously bad problem or solution you have encountered in school?
94 replies
1 viewing
LXC007
May 21, 2025
cheltstudent
20 minutes ago
J
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Grandi Series is weird??
sadas123   6
N 22 minutes ago by cheltstudent
I was just watching this video and I kinda understood what they were saying they said:

$1-1+1-1+1-1+1-1 .... = \frac{1}{2}$

so at first I thought that = 0

but then I realized that if you give an variable such that:

$x= 1-1+1-1+1-1 .....$
then you get
$x = 1-x$
isolating x you get that
$2x = 1$
so
$x= \frac{1}{2}$

is there a way to get the definite answer because I am still confused.
6 replies
sadas123
2 hours ago
cheltstudent
22 minutes ago
J
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Creative Geometry Problem
giratina3   2
N 24 minutes ago by giratina3
Let there be a triangle ABC. Let BX bisect CBA, CY bisect ACB, and XY parallel to BC. If AB = 12, BC = 24, and AC = 18, then what is the perimeter of the triangle AXY?

The hint gives away most of the problem, so make sure you consider the problem for a long time before reading it :thumbup:

(The answer is an integer to you coordinate bashers :play_ball:)

Hint
2 replies
giratina3
Yesterday at 8:10 PM
giratina3
24 minutes ago
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Is it true?
lgx57   1
N 5 hours ago by alexheinis
$0<a_1,a_2\cdots ,a_n$, determine whether it is true.
$$\sum_{i=1}^n \frac{1}{a_i}\ge \sum_{i=1}^n \frac{i}{\sum_{j=1}^i a_j}$$
If not, please give a counterexample.
1 reply
lgx57
Yesterday at 3:02 PM
alexheinis
5 hours ago
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Cone Sul 2020 TST 3 Brazil P2
TiagoCamara   0
6 hours ago
(Cone Sul 2020 TST 3 Brazil P2)Determine all positive integers $n$ for which $4k^2+n$ is a prime number for every $0\leq k< n$ integer.
0 replies
TiagoCamara
6 hours ago
0 replies
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Limit of a sequence involving the largest odd divisor
JackMinhHieu   1
N Yesterday at 7:15 PM by mathreyes
Hi everyone,

I came across the following sequence and I’m curious about its behavior:

Let d(k) be the largest odd positive divisor of k. Define a sequence (x_n) by

x_n = (1/n) * sum_{k=1}^{n} (d(k)/k)

Question:
Does the sequence (x_n) converge? If so, what is its limit?

Any insights, proofs, or helpful observations would be appreciated. Thank you!
1 reply
1 viewing
JackMinhHieu
Yesterday at 5:19 PM
mathreyes
Yesterday at 7:15 PM
J
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Chinese Remainder Theorem
MathNerdRabbit103   0
Yesterday at 6:19 PM
Hi guys,
Lately i've been trying to understand the proof for the Chinese Remainder Theorem, however i have unfortunately had no luck. Can anybody post about how they understand the proof and please go step by step?
Appreciate it.
0 replies
MathNerdRabbit103
Yesterday at 6:19 PM
0 replies
J
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An Angle Trisector
bryanguo   3
N Yesterday at 5:16 PM by Sedro
Triangle $ABC$ has points $D$,$E$,$F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$, and $AD$ and $AE$ trisect angle $BAF$. If $\angle BAF = 60^{\circ}$, $\frac{EF}{EC}=\frac{2}{3}$, and $\frac{AE}{AC} = 2$, find $\angle BAC$.

Individual #5
3 replies
bryanguo
Apr 11, 2024
Sedro
Yesterday at 5:16 PM
J
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10 Problems
Sedro   4
N Yesterday at 4:49 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 $995$.
4 replies
Sedro
Jul 10, 2025
Sedro
Yesterday at 4:49 PM
J
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Find $a, b$ are positive integers such that $a\le2018$ and $a^5+b^7\vdots 2018
Limited1   3
N Yesterday at 2:56 PM by boylearnmath
Let $a,b$ be positive integers such that $a\le2018$ and $$a^5+b^7\vdots 2018$$. Find $a$
3 replies
Limited1
Aug 14, 2022
boylearnmath
Yesterday at 2:56 PM
J
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Real numbers
lskgmkj   0
Yesterday at 2:55 PM

[ x - (- y) ] = +z
now, here when we subtract negative integer from positive integer why we add the numbers i am not talking ABOUT BASIC THING but can you explain it with proof and logically not like middle school maths.
i would be grateful for your help given
0 replies
lskgmkj
Yesterday at 2:55 PM
0 replies
J
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Geometry easy
AlexCenteno2007   3
N Yesterday at 2:15 PM by mathprodigy2011
In triangle ABC, if angle B=120°, AB=5u and BC=15u. Draw the interior bisector BE. Calculate BE
3 replies
AlexCenteno2007
Friday at 10:55 PM
mathprodigy2011
Yesterday at 2:15 PM
J
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[20th PMO Area Stage: I. 5]
reilynso   1
N Yesterday at 10:21 AM by P0tat0b0y
Let $f(x)=\sqrt{4 \sin^4 x - \sin ^2 x \cos ^2 x + 4 \cos^4 x}$ for any $x \in \mathbb{R}$. Let $M$ and $m$ be the maximum and minimum values of $f$, respectively. Find the product of $M$ and $m$.

Answer

Solution
1 reply
1 viewing
reilynso
Yesterday at 9:11 AM
P0tat0b0y
Yesterday at 10:21 AM
J
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J
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Need some help...
Rice_Farmer   3
N Dec 1, 2023 by Zionqupaul
---------------------------------------------
$25$ books of the same kind are divided into $6$ groups. Each group has at least one book and each group has a different number of books. How many ways are there to divide the books?
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3 replies
Rice_Farmer
Nov 30, 2023
Zionqupaul
Dec 1, 2023
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Need some help...
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Reveal topic tags
Counting and Probability
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Rice_Farmer
961 posts
Rice_Farmer
#1 h  Nov 30, 2023, 10:54 PM
Y by
$25$ books of the same kind are divided into $6$ groups. Each group has at least one book and each group has a different number of books. How many ways are there to divide the books?
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lithia5458
39 posts
lithia5458
#2 h  Nov 30, 2023, 11:00 PM
Y by
To start, try making a table with a descending order of numbers (example: 3, 2, 1).
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Rice_Farmer
961 posts
Rice_Farmer
#3 h  Nov 30, 2023, 11:11 PM
Y by
lithia5458 wrote:
To start, try making a table with a descending order of numbers (example: 3, 2, 1).

Yes, I tried that

I ended up brute forcing it and getting $5.$

The book solution made no sense and I would like a better solution than just brute force
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Zionqupaul
494 posts
Zionqupaul
#4 h  Dec 1, 2023, 1:10 AM
Y by
Minimum number of books: 1+2+3+4+5+6=21, increase by 4. In order to divide the books, you must increase the largest group first
Do case work:
Case 1: only increase the group of 6 to reach 28
case 2: only increase the groups of 5,6 to reach 28
and continue
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