Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Happy Memorial Day! Please note that AoPS Online is closed May 24-26th.
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
J
H
22nd PMO, Qualifying Stage II.6
elpianista227   1
N 20 minutes ago by elpianista227
Find the sum of all real numbers $b$ for which the roots of the equation $x^2 + bx - 3b = 0$ are integers.
1 reply
elpianista227
29 minutes ago
elpianista227
20 minutes ago
J
H
[PMO 17] Area Stage I. #14
NeoAzure   0
43 minutes ago
14. In how many ways can Alex, Billy, and Charles split 7 identical marbles among themselves
so that no two have the same number of marbles? It is possible for someone not to get any
marbles.

Answer
0 replies
1 viewing
NeoAzure
43 minutes ago
0 replies
J
H
[18th PMO Area Stage I. #4] logs + geo
ACalculationError   1
N an hour ago by ACalculationError
Let \( f(x) = \log_a x \) for some base \( a > 0 \), \( a \neq 1 \). The points \( (3, m) \), \( (x_1, y_1) \), and \( (x_2, y_2) \) lie on the graph of \( f \). If \(y_1 + y_2 = 2m\), find the value of \( x_1 x_2 \).

Answer Confirmation
1 reply
ACalculationError
an hour ago
ACalculationError
an hour ago
J
H
a tst 2013 test
Math2030   1
N 2 hours ago by Math2030
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
1 reply
Math2030
Today at 5:26 AM
Math2030
2 hours ago
J
H
[Sipnayan JHS] Semifinals Round B, Average, #2
LilKirb   1
N 3 hours ago by LilKirb
How many trailing zeroes are there in the base $4$ representation of $2015!$ ?
1 reply
LilKirb
4 hours ago
LilKirb
3 hours ago
J
H
2022 SMT Team Round - Stanford Math Tournament
parmenides51   5
N 4 hours ago by vanstraelen
p1. Square $ABCD$ has side length $2$. Let the midpoint of $BC$ be $E$. What is the area of the overlapping region between the circle centered at $E$ with radius $1$ and the circle centered at $D$ with radius $2$? (You may express your answer using inverse trigonometry functions of noncommon values.)


p2. Find the number of times $f(x) = 2$ occurs when $0 \le x \le 2022 \pi$ for the function $f(x) = 2^x(cos(x) + 1)$.


p3. Stanford is building a new dorm for students, and they are looking to offer $2$ room configurations:
$\bullet$ Configuration $A$: a one-room double, which is a square with side length of $x$,
$\bullet$ Configuration $B$: a two-room double, which is two connected rooms, each of them squares with a side length of $y$.
To make things fair for everyone, Stanford wants a one-room double (rooms of configuration $A$) to be exactly $1$ m$^2$ larger than the total area of a two-room double. Find the number of possible pairs of side lengths $(x, y)$, where $x \in N$, $y \in N$, such that $x - y < 2022$.


p4. The island nation of Ur is comprised of $6$ islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands $A$ and $B$ if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into?


p5. Let $a, b,$ and $c$ be the roots of the polynomial $x^3 - 3x^2 - 4x + 5$. Compute $\frac{a^4 + b^4}{a + b}+\frac{b^4 + c^4}{b + c}+\frac{c^4 + a^4}{c + a}$.


p6. Carol writes a program that finds all paths on an 10 by 2 grid from cell (1, 1) to cell (10, 2) subject to the conditions that a path does not visit any cell more than once and at each step the path can go up, down, left, or right from the current cell, excluding moves that would make the path leave the grid. What is the total length of all such paths? (The length of a path is the number of cells it passes through, including the starting and ending cells.)


p7. Consider the sequence of integers an defined by $a_1 = 1$, $a_p = p$ for prime $p$ and $a_{mn} = ma_n + na_m$ for $m, n > 1$. Find the smallest $n$ such that $\frac{a_n^2}{2022}$ is a perfect power of $3$.


p8. Let $\vartriangle ABC$ be a triangle whose $A$-excircle, $B$-excircle, and $C$-excircle have radii $R_A$, $R_B$, and $R_C$, respectively. If $R_AR_BR_C = 384$ and the perimeter of $\vartriangle ABC$ is $32$, what is the area of $\vartriangle ABC$?


p9. Consider the set $S$ of functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 243\}$ satisfying:
(a) $f(1) = 1$
(b) $f(n^2) = n^2f(n)$,
(c) $n |f(n)$,
(d) $f(lcm(m, n))f(gcd(m, n)) = f(m)f(n)$.
If $|S|$ can be written as $p^{\ell_1}_1 \cdot p^{\ell_2}_2 \cdot ... \cdot p^{\ell_k}_k$ where $p_i$ are distinct primes, compute $p_1\ell_1+p_2\ell_2+. . .+p_k\ell_k$.


p10. You are given that $\log_{10}2 \approx 0.3010$ and that the first (leftmost) two digits of $2^{1000}$ are 10. Compute the number of integers $n$ with $1000 \le n \le 2000$ such that $2^n$ starts with either the digit $8$ or $9$ (in base $10$).


p11. Let $O$ be the circumcenter of $\vartriangle ABC$. Let $M$ be the midpoint of $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively, onto the opposite sides. $EF$ intersects $BC$ at $P$. The line passing through $O$ and perpendicular to $BC$ intersects the circumcircle of $\vartriangle ABC$ at $L$ (on the major arc $BC$) and $N$, and intersects $BC$ at $M$. Point $Q$ lies on the line $LA$ such that $OQ$ is perpendicular to $AP$. Given that $\angle BAC = 60^o$ and $\angle AMC = 60^o$, compute $OQ/AP$.


p12. Let $T$ be the isosceles triangle with side lengths $5, 5, 6$. Arpit and Katherine simultaneously choose points $A$ and $K$ within this triangle, and compute $d(A, K)$, the squared distance between the two points. Suppose that Arpit chooses a random point $A$ within $T$ . Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of $d(A, K)$. Compute this value.


p13. For a regular polygon $S$ with $n$ sides, let $f(S)$ denote the regular polygon with $2n$ sides such that the vertices of $S$ are the midpoints of every other side of $f(S)$. Let $f^{(k)}(S)$ denote the polygon that results after applying f a total of k times. The area of $\lim_{k \to \infty} f^{(k)}(P)$ where $P$ is a pentagon of side length $1$, can be expressed as $\frac{a+b\sqrt{c}}{d}\pi^m$ for some positive integers $a, b, c, d, m$ where $d$ is not divisible by the square of any prime and $d$ does not share any positive divisors with $a$ and $b$. Find $a + b + c + d + m$.


p14. Consider the function $f(m) = \sum_{n=0}^{\infty}\frac{(n - m)^2}{(2n)!}$ . This function can be expressed in the form $f(m) = \frac{a_m}{e} +\frac{b_m}{4}e$ for sequences of integers $\{a_m\}_{m\ge 1}$, $\{b_m\}_{m\ge 1}$. Determine $\lim_{n \to \infty}\frac{2022b_m}{a_m}$.


p15. In $\vartriangle ABC$, let $G$ be the centroid and let the circumcenters of $\vartriangle BCG$, $\vartriangle CAG$, and $\vartriangle ABG$ be $I, J$, and $K$, respectively. The line passing through $I$ and the midpoint of $BC$ intersects $KJ$ at $Y$. If the radius of circle $K$ is $5$, the radius of circle $J$ is $8$, and $AG = 6$, what is the length of $KY$ ?



PS. You should use hide for answers. Collected here.
5 replies
parmenides51
Jun 30, 2022
vanstraelen
4 hours ago
J
H
[Sipnayan SHS] Finals Round, Difficult
LilKirb   1
N Today at 6:59 AM by LilKirb
Let $f$ be a polynomial with nonnegative integer coefficients. If $f(1) = 11$ and $f(11) = 2311$, what is the remainder when $f(10)$ is divided by $1000?$
1 reply
LilKirb
Today at 6:49 AM
LilKirb
Today at 6:59 AM
J
H
Inequalities
sqing   1
N Today at 3:50 AM by sqing
Let $ a,b> 0 ,\frac{a}{2b+1}+\frac{b}{3}+\frac{1}{2a+1} \leq 1.$ Prove that
$$ a^2+b^2 -ab\leq 1$$$$ a^2+b^2 +ab \leq3$$Let $ a,b,c> 0 , \frac{a}{2b+1}+\frac{b}{2c+1}+\frac{c}{2a+1} \leq 1.$ Prove that
$$ a +b +c +abc \leq 4$$
1 reply
sqing
Today at 3:11 AM
sqing
Today at 3:50 AM
J
H
Inequalities
sqing   19
N Today at 2:50 AM by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
19 replies
sqing
May 13, 2025
sqing
Today at 2:50 AM
J
H
inequality
luckvoltia.112   9
N Today at 2:35 AM by Shan3t
Given that \( a, b, c, d \) are nonzero real numbers, find the minimum value of the expression
\[ P = \left| \frac{b + c + d}{a} \right| + \left| \frac{c + d + a}{b} \right| + \left| \frac{d + a + b}{c} \right| + \left| \frac{a + b + c}{d} \right|. \]
9 replies
luckvoltia.112
Today at 12:45 AM
Shan3t
Today at 2:35 AM
J
H
prove that
luckvoltia.112   0
Today at 12:52 AM
Let \( a, b, c \) be non-negative real numbers such that \( a + b + c > 0 \) and
\[ \frac{25a + 36b + 49c}{5a + 6b + 7c} + \frac{25b + 36c + 49a}{5b + 6c + 7a} + \frac{25c + 36a + 49b}{5c + 6a + 7b} = 18. \]Prove that exactly two of the numbers \( a, b, c \) are equal to 0.
0 replies
luckvoltia.112
Today at 12:52 AM
0 replies
J
H
Geometry Trigonometry Olympiads
Foxellar   0
Yesterday at 11:07 PM
Let \( \triangle ABC \) be a triangle such that \( \angle ABC = 120^\circ \). Points \( X, Y, Z \) lie on segments \( BC, CA, AB \), respectively, such that lines \( AX, BY, \) and \( CZ \) are the angle bisectors of triangle \( ABC \). Find the measure of angle \( \angle XYZ \).
0 replies
Foxellar
Yesterday at 11:07 PM
0 replies
J
H
Proof of ramsey number
smadadi1000   1
N Yesterday at 10:35 PM by smadadi1000
How do you prove that r(n,2)=n using the pigeonhole principle?
1 reply
smadadi1000
Yesterday at 10:31 PM
smadadi1000
Yesterday at 10:35 PM
J
H
Minimize
lgx57   1
N Yesterday at 5:25 PM by Math-lover1
Minimize $\sqrt{\cos^2 x+(2-\sin x)^2}+\dfrac{1}{2}\sqrt{(\sqrt 3-\cos x)^2+(\sin x+1)^2}$
1 reply
lgx57
Yesterday at 1:29 PM
Math-lover1
Yesterday at 5:25 PM
J
H
J
H
Sequence
lgx57   8
N Apr 30, 2025 by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Apr 30, 2025
J
Sequence
G H J
Reveal topic tags
algebra
Sequences
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
44 posts
lgx57
#1 h  Apr 27, 2025, 12:56 PM
Y by
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
44 posts
lgx57
#2 h  Apr 27, 2025, 12:57 PM
Y by
I can only find that $a_n \sim \sqrt{2n}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aops-g5-gethsemanea2
3458 posts
aops-g5-gethsemanea2
#3 h  Apr 27, 2025, 2:05 PM
Y by
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
44 posts
lgx57
#4 h  Apr 27, 2025, 2:19 PM
Y by
aops-g5-gethsemanea2 wrote:
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?

Just find a function $f$ ,s.t. $a_n=f(n)$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
steve4916
64 posts
steve4916
#5 h  Apr 27, 2025, 2:24 PM
Y by
now prove me if im wrong but there is no simple closed form for this
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
44 posts
lgx57
#6 h  Apr 27, 2025, 2:48 PM
Y by
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
johnnie.walker
2 posts
johnnie.walker
#7 h  Apr 29, 2025, 1:42 PM
Y by
Click to reveal hidden text
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jasperE3
11379 posts
jasperE3
#8 h  Apr 29, 2025, 3:38 PM
Y by
lgx57 wrote:
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?

why would there be a closed form
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vivaandax
87 posts
Vivaandax
#9 h  Apr 30, 2025, 5:08 AM
Y by
You can bound the value of a_n quite well (consider IMO Shortlist 1975 Problem 14), but there is not an explicit formula to calculate a_n.
Z K Y
N Quick Reply
G
H
=
a