Have a great winter break! Please note that AoPS Online will not have classes Dec 21, 2024 through Jan 3, 2025.

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k a December Highlights and 2024 AoPS Online Class Information
jlacosta   0
Dec 2, 2024
Warmest Holiday Wishes!
As 2024 draws to a close and we look ahead to next year, we find ourselves in a perfect moment for reflection. This season gives us a chance to celebrate our successes, learn from our challenges, and express gratitude to everyone who has been part of our problem-solving journey.

At AoPS, we're especially thankful for students like you who share our passion for tackling interesting problems and discovering inspiration and joy in the process. Your enthusiasm drives us to keep improving and growing. May you and yours have a wonderful winter break and holiday season.

Be sure to mark your calendars for these upcoming dates:

[list][*]It’s not too late to train for AMC 8 with our accelerated sections that finish before the exam takes place between January 22 - January 28. Our AMC 8/MATHCOUNTS Basics accelerated section starts on December 2nd and AMC 8/MATHCOUNTS Advanced has two accelerated classes, one begins on December 2nd and the other December 10th.
[*]Crack the code for how to participate in your first USA Computing Olympiad at our FREE math jam on December 10th.
[*]It’s all relative! Join our Relativity weekend seminar December 14 - December 15 and explore Einstein's theory of special relativity.
[*]Enroll soon if you want to start a class in December! We have a number of options for introductory, intermediate, and advanced courses that begin December 2 - 11.
[*]Please note: AoPS will not be holding classes December 21 ‐ January 3 and our office will be closed December 24 - January 3.[/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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0 replies
jlacosta
Dec 2, 2024
0 replies
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
2013-2014 Fall OMO #24
v_Enhance   18
N Jul 22, 2024 by dudade
The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy $a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}$ and $b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}$ for every integer $n = 1, 2, \dots, 2013$. If $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute \[ \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). \]Proposed by Evan Chen
18 replies
v_Enhance
Oct 30, 2013
dudade
Jul 22, 2024
2014-2015 Fall OMO #13
v_Enhance   3
N Jun 29, 2024 by Deelz
Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test has $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat gets the $n$th problem correct with probability $\frac{1}{n^2}$ while Q gets the $n$th problem correct with probability $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly.

Just before turning in their test, the ducks and ducklings decide to share answers! On any question which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores?

Proposed by Evan Chen
3 replies
v_Enhance
Oct 28, 2014
Deelz
Jun 29, 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
2013-2014 Fall OMO #26
v_Enhance   7
N Mar 11, 2024 by HamstPan38825
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
7 replies
v_Enhance
Oct 30, 2013
HamstPan38825
Mar 11, 2024
2013-2014 Spring OMO #25
v_Enhance   11
N Jan 22, 2024 by HamstPan38825
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.

Proposed by Michael Kural
11 replies
v_Enhance
Apr 15, 2014
HamstPan38825
Jan 22, 2024
2012-2013 Winter OMO #29
v_Enhance   7
N Jan 16, 2024 by shendrew7
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$, and let $d(n)$ denote the number of positive integer divisors of $n$. For example, $\phi(6) = 2$ and $d(6) = 4$. Find the sum of all odd integers $n \le 5000$ such that $n \mid \phi(n) d(n)$.

Alex Zhu
7 replies
v_Enhance
Jan 16, 2013
shendrew7
Jan 16, 2024
OMO Fall 2013 Solutions
v_Enhance   4
N Nov 20, 2013 by v_Enhance
Took long enough. Here you are!
4 replies
v_Enhance
Nov 19, 2013
v_Enhance
Nov 20, 2013
OMO Fall 2013 Solutions
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G H BBookmark kLocked kLocked NReply
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v_Enhance
6845 posts
#1 • 7 Y
Y by El_Ectric, djmathman, ahaanomegas, forthegreatergood, minimario, math154, Adventure10
Took long enough. Here you are!
Attachments:
OMOFall13Solns.pdf (311kb)
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El_Ectric
3091 posts
#2 • 2 Y
Y by Adventure10, Mango247
Thanks! Just FYI, the diagram for #9 is misplaced. EDIT: Also the diagram for #25.

EDIT 2: the "this RSI 2013 paper" link in the solution to #27 doesn't work. ("Nothing here")
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v_Enhance
6845 posts
#3 • 2 Y
Y by Adventure10, Mango247
Oops, forgot that I moved the RSI paper somewhere else. The new link should work.

The diagrams moving around is because LaTeX floats figures; LaTeX does its best that it can to find a reasonable location to place them.
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Rama
208 posts
#4 • 2 Y
Y by Adventure10, Mango247
Can someone elaborate on the comment for #1? I don't exactly see why the cycle shifting follows from $1/7=0.\overline{142857}$ and that 10 is a primitive root modulo 7.
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v_Enhance
6845 posts
#5 • 5 Y
Y by ahaanomegas, Rama, happiface, Adventure10, Mango247
\begin{align*}
		0.\overline{142857} &= \frac{1}{7} = \frac{10^{0}}{7} - 0 \\
		0.\overline{285714} &= \frac{2}{7} = \frac{10^{2}}{7} - 14 \\
		0.\overline{428571} &= \frac{3}{7} = \frac{10^{1}}{7} - 1 \\
		0.\overline{571428} &= \frac{4}{7} = \frac{10^{4}}{7} - 1428 \\
		0.\overline{714285} &= \frac{5}{7} = \frac{10^{5}}{7} - 14285 \\
		0.\overline{857142} &= \frac{6}{7} = \frac{10^{3}}{7} - 142
	\end{align*}
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