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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
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0 replies
jlacosta
Mar 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
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
2017-2018 Fall OMO Problem 18
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trumpeter
3332 posts
#1 • 1 Y
Y by Adventure10
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
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niyu
830 posts
#2 • 4 Y
Y by Math1331Math, Benq, The_Maitreyo1, Adventure10
Solution
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AOPS12142015
1448 posts
#5 • 2 Y
Y by Adventure10, Mango247
Calculus
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whatshisbucket
975 posts
#6 • 2 Y
Y by Makorn, Adventure10
Solution:
The answer is an integer.
$(3,2,7)$ is a solution to both equations in the integers.
This works better than other permutations of those 3 numbers
Therefore, the answer is $56.$
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samuel
442 posts
#7 • 2 Y
Y by Adventure10, Mango247
whatshisbucket wrote:
Solution:
The answer is an integer.
$(3,2,7)$ is a solution to both equations in the integers.
This works better than other permutations of those 3 numbers
Therefore, the answer is $56.$
I did it the same way :P
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DaniyalQazi2
757 posts
#8 • 2 Y
Y by Adventure10, Mango247
I have a really bad solution with a lot of assumptions:
Bad Solution with lots of assumptions
Great problem!
This post has been edited 2 times. Last edited by DaniyalQazi2, Nov 8, 2017, 4:05 AM
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ryanbear
1055 posts
#9
Y by
$abc-(a+2b-3c)=ab+bc+ac+1-(a+2b-3c)=\frac{(a+b+c)^2-a^2-b^2-c^2+2-2a-4b+6c}{2}=\frac{146-a^2-b^2-c^2-2a-4b+6c}{2}=\frac{160-(a+1)^2-(b+2)^2-(c-3)^2}{2}$. Note that to minimize this, $a+1=b+2=c-3$. So $(a,b,c)=(3,2,7)$ works. This results in the answer being $\boxed{56}.$
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