It's February and we'd love to help you find the right course plan!

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k a February Highlights and 2025 AoPS Online Class Information
jlacosta   0
Feb 2, 2025
We love to share what you can look forward to this month! The AIME I and AIME II competitions are happening on February 6th and 12th, respectively. Join our Math Jams the day after each competition where we will go over all the problems and the useful strategies to solve them!

2025 AIME I Math Jam: Difficulty Level: 8* (Advanced math)
February 7th (Friday), 4:30pm PT/7:30 pm ET

2025 AIME II Math Jam: Difficulty Level: 8* (Advanced math)
February 13th (Thursday), 4:30pm PT/7:30 pm ET

The F=ma exam will be held on February 12th. Check out our F=ma Problem Series course that begins February 19th if you are interested in participating next year! The course will prepare you to take the F=ma exam, the first test in a series of contests that determines the members of the US team for the International Physics Olympiad. You'll learn the classical mechanics needed for the F=ma exam as well as how to solve problems taken from past exams, strategies to succeed, and you’ll take a practice F=ma test of brand-new problems.

Mark your calendars for all our upcoming events:
[list][*]Feb 7, 4:30 pm PT/7:30pm ET, 2025 AIME I Math Jam
[*]Feb 12, 4pm PT/7pm ET, Mastering Language Arts Through Problem-Solving: The AoPS Method
[*]Feb 13, 4:30 pm PT/7:30pm ET, 2025 AIME II Math Jam
[*]Feb 20, 4pm PT/7pm ET, The Virtual Campus Spring Experience[/list]
AoPS Spring classes are open for enrollment. Get a jump on 2025 and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
Many of you are beginning to build your education and achievements history on LinkedIn. Now, you can showcase your courses from Art of Problem Solving (AoPS) directly on your LinkedIn profile! Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

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.

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Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

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Sat & Sun, Feb 1 - Feb 2 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

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Wednesday, Feb 19 - May 7

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0 replies
jlacosta
Feb 2, 2025
0 replies
2014-2015 Fall OMO #26
v_Enhance   14
N Feb 9, 2025 by abeot
Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$.

Proposed by Michael Kural
14 replies
v_Enhance
Oct 28, 2014
abeot
Feb 9, 2025
2018-2019 Fall OMO Problem 30
trumpeter   17
N Jan 10, 2025 by qwerty123456asdfgzxcvb
Let $ABC$ be an acute triangle with $\cos B =\frac{1}{3}, \cos C =\frac{1}{4}$, and circumradius $72$. Let $ABC$ have circumcenter $O$, symmedian point $K$, and nine-point center $N$. Consider all non-degenerate hyperbolas $\mathcal H$ with perpendicular asymptotes passing through $A,B,C$. Of these $\mathcal H$, exactly one has the property that there exists a point $P\in \mathcal H$ such that $NP$ is tangent to $\mathcal H$ and $P\in OK$. Let $N'$ be the reflection of $N$ over $BC$. If $AK$ meets $PN'$ at $Q$, then the length of $PQ$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers such that $c$ is not divisible by the square of any prime. Compute $100a+b+c$.

Proposed by Vincent Huang
17 replies
trumpeter
Nov 7, 2018
qwerty123456asdfgzxcvb
Jan 10, 2025
2013-2014 Fall OMO #26
v_Enhance   8
N Jan 8, 2025 by OronSH
Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan  A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$.

Proposed by Michael Kural
8 replies
v_Enhance
Oct 30, 2013
OronSH
Jan 8, 2025
2012-2013 Winter OMO #22
v_Enhance   2
N Dec 28, 2024 by NicoN9
In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$.

Ray Li
2 replies
v_Enhance
Jan 16, 2013
NicoN9
Dec 28, 2024
2012-2013 Winter OMO #11
v_Enhance   2
N Dec 26, 2024 by NicoN9
Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$?

Ray Li
2 replies
v_Enhance
Jan 16, 2013
NicoN9
Dec 26, 2024
2011-2012 Winter OMO #22
Zhero   4
N Dec 24, 2024 by NicoN9
Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.

Author: Alex Zhu
4 replies
Zhero
Jan 24, 2012
NicoN9
Dec 24, 2024
2013-2014 Fall OMO #29
v_Enhance   22
N Dec 11, 2024 by eg4334
Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Proposed by Ray Li
22 replies
v_Enhance
Oct 30, 2013
eg4334
Dec 11, 2024
2012-2013 Winter OMO #38
v_Enhance   11
N Oct 29, 2024 by PEKKA
Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$.

Victor Wang
11 replies
v_Enhance
Jan 16, 2013
PEKKA
Oct 29, 2024
2015-2016 Fall OMO #12
pi37   14
N Aug 7, 2024 by eg4334
Let $a$, $b$, $c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$.
The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2$, $ca-b^2$, $ab-c^2$.
What is the sum of the coefficients of $Q$?

Proposed by Evan Chen
14 replies
pi37
Nov 18, 2015
eg4334
Aug 7, 2024
2017-2018 Fall OMO Problem 18
trumpeter   6
N Aug 5, 2024 by ryanbear
Let $a,b,c$ be real nonzero numbers such that $a+b+c=12$ and \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=1.\]Compute the largest possible value of $abc-\left(a+2b-3c\right)$.

Proposed by Tristan Shin
6 replies
trumpeter
Nov 7, 2017
ryanbear
Aug 5, 2024
2015-2016 Spring OMO #17
mathocean97   9
N May 2, 2024 by Brudder
A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

(a) If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
(b) If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b\le 50$ such that if $a, b \in S$ then $S = \mathbb{N}.$

Proposed by Yang Liu
9 replies
mathocean97
Mar 29, 2016
Brudder
May 2, 2024
2015-2016 Spring OMO #17
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mathocean97
606 posts
#1 • 2 Y
Y by Adventure10, Mango247
A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

(a) If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
(b) If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b\le 50$ such that if $a, b \in S$ then $S = \mathbb{N}.$

Proposed by Yang Liu
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atmchallenge
980 posts
#2 • 2 Y
Y by mathmonster369, Adventure10
Brute Force
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Tommy2000
715 posts
#3 • 2 Y
Y by Adventure10, Mango247
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somepersonoverhere
1313 posts
#4 • 2 Y
Y by Adventure10, Mango247
$$50 \times 50 - \lfloor \frac{50}{3} \rfloor^2 - \lfloor \frac{50}{5} \rfloor^2 - ..... - \lfloor \frac{50}{47} \rfloor^2 + \lfloor \frac{50}{15} \rfloor^2 + \lfloor \frac{50}{21} \rfloor^2 + ... + \lfloor \frac{50}{39} \rfloor^2 = 2068$$after proving some stuff with chicken mcnugget theorem
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champion999
1530 posts
#5 • 2 Y
Y by Adventure10, Mango247
Ugh got $2052$, messed up my computation and dropped a $16$ somewhere.
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Shaddoll
688 posts
#6 • 2 Y
Y by Adventure10, Mango247
Solution outline.
This post has been edited 1 time. Last edited by Shaddoll, Mar 30, 2016, 12:59 AM
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MSTang
6012 posts
#7 • 1 Y
Y by Adventure10
Another way to prove that $(a, b)$ works, where $a$ and $b$ are odd and relatively prime, is to use strong induction on the sum $a+b$.
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acegikmoqsuwy2000
767 posts
#8 • 2 Y
Y by Adventure10, Mango247
are you kidding me, i think my team may have put 2069
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Darn
996 posts
#10 • 2 Y
Y by Adventure10, Mango247
Our team used totients instead to calculate the number of pairs. I think this was slightly cleaner than the PIE bash.

Once you reduce the problem to finding ordered pairs $(a,b) \in \{1,2,3,\ldots,50\}^2$ for which $\gcd(a,b)=2^{k}$ where $k$ is a nonnegative integer, you can express the total as \[ 2\left(\sum\limits_{k=2}^{50} \phi(k)\right) + 2\left(\sum\limits_{k=2}^{25} \phi(k)\right)+2\left(\sum\limits_{k=2}^{12} \phi(k)\right) +2\left(\sum\limits_{k=2}^{6} \phi(k)\right)+2\left(\sum\limits_{k=2}^{3} \phi(k)\right)+6.\]An explanation of where the sum comes from:

Now the problem reduces to finding totients from $2$ to $50$, which is really not bad at all.
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Brudder
416 posts
#11
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If we fix $m = \frac{50}{2^k}$ for some $k,$ then we can count the number of pairs having $\gcd(a, b) = 2^k$ as

\begin{align}
&\sum_{a = 1}^{m} \sum_{b=1}^{m} [\gcd(a, b) = 1] \\&= 
\sum_{a=1}^m \sum_{b=1}^m \sum_{d | \gcd(a, b)} \mu(d) \\&=
\sum_{a=1}^m \sum_{b=1}^m \sum_{d=1}^m [d|a][d|b] \mu(d) \\&= 
\sum_{d=1}^m \mu(d) \sum_{a=1}^m [d|a] \sum_{b=1}^m [d|b] \\&=
\sum_{d=1}^m \mu(d) \left(\lfloor \frac{m}{d} \rfloor\right)^2
\end{align}
We can compute this quantity for $k = 0, 1, \dots, 5$ which yields the correct answer. Note that we can group terms together that have common $\lfloor \frac{m}{d} \rfloor$ values, significantly reducing computation
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