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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
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
a