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The OMO, a series of free team-based math competitions
The OMO, a series of free team-based math competitions
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MG
Topic
First Poster
Last Poster
2014-2015 Fall OMO #26
v_Enhance 14
N
Feb 9, 2025
by abeot
Let
be a triangle with
,
,
. Let
,
,
be the midpoints of arcs
,
,
(not containing the opposite vertices) respectively on the circumcircle of
. Let
be the midpoint of arc
containing point
. Suppose lines
and
meet at
, while lines
and
meet at
. Find the square of the distance from
to
.
Proposed by Michael Kural






















Proposed by Michael Kural
14 replies
2013-2014 Fall OMO #26
v_Enhance 8
N
Jan 8, 2025
by OronSH
Let
be a triangle with
,
, and
. Denote the reflections of
across
by
, respectively, and let
be the circumcenter of triangle
. Let
be a point such that
, and let
and
be the midpoints of the major and minor arcs
of the circumcircle of triangle
. Find
.
Proposed by Michael Kural
















Proposed by Michael Kural
8 replies
2012-2013 Winter OMO #22
v_Enhance 2
N
Dec 28, 2024
by NicoN9
In triangle
,
,
, and
. Let
be the point on segment
satisfying
, and let
be the unique point such that
and line
is tangent to the circumcircle of
. Find the length of segment
.
Ray Li












Ray Li
2 replies
2012-2013 Winter OMO #11
v_Enhance 2
N
Dec 26, 2024
by NicoN9
Let
,
, and
be distinct points on a line with
. Square
and equilateral triangle
are drawn on the same side of line
. What is the degree measure of the acute angle formed by lines
and
?
Ray Li









Ray Li
2 replies
2011-2012 Winter OMO #22
Zhero 4
N
Dec 24, 2024
by NicoN9
Find the largest prime number
such that when
is written in base
, it has at least
trailing zeroes.
Author: Alex Zhu




Author: Alex Zhu
4 replies
2013-2014 Fall OMO #29
v_Enhance 22
N
Dec 11, 2024
by eg4334
Kevin has
cookies, each labeled with a unique nonempty subset of
. 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
, he eats that cookie as well as the cookies with
and
). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Proposed by Ray Li









Proposed by Ray Li
22 replies
2012-2013 Winter OMO #38
v_Enhance 11
N
Oct 29, 2024
by PEKKA
Triangle
has sides
,
, and
. Let
be the points on segments
, respectively, such that
and
. Suppose lines
and
intersect at
and the circumcircles of
and
intersect at a second point
. If the length of segment
can be expressed in the form
for positive integers
, where
is not divisible by the square of any prime, find
.
Victor Wang



















Victor Wang
11 replies
2015-2016 Fall OMO #12
pi37 14
N
Aug 7, 2024
by eg4334
Let
,
,
be the distinct roots of the polynomial
.
The cubic polynomial
is monic and has distinct roots
,
,
.
What is the sum of the coefficients of
?
Proposed by Evan Chen




The cubic polynomial




What is the sum of the coefficients of

Proposed by Evan Chen
14 replies
