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USAM(inimize)OOO
277546   72
N Yesterday at 1:57 AM by Maximilian113
Source: 2020 USOMO Problem 1
Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.

Proposed by Zuming Feng
72 replies
277546
Jun 21, 2020
Maximilian113
Yesterday at 1:57 AM
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Geo is back??
GoodMorning   136
N Tuesday at 11:17 PM by eg4334
Source: 2023 USAJMO Problem 2/USAMO Problem 1
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.

Proposed by Holden Mui
136 replies
GoodMorning
Mar 23, 2023
eg4334
Tuesday at 11:17 PM
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memorize your 60 120 degree triangles
OronSH   22
N Apr 7, 2025 by xHypotenuse
Source: 2024 AMC 12A #19
Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$ and $DA=5$ with $\angle CDA=120^\circ$. What is the length of the shorter diagonal of $ABCD$?

$ \textbf{(A) }\frac{31}7 \qquad \textbf{(B) }\frac{33}7 \qquad \textbf{(C) }5 \qquad \textbf{(D) }\frac{39}7 \qquad \textbf{(E) }\frac{41}7 \qquad $
22 replies
OronSH
Nov 7, 2024
xHypotenuse
Apr 7, 2025
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usamOOK geometry
KevinYang2.71   94
N Apr 7, 2025 by SatisfiedMagma
Source: USAMO 2025/4, USAJMO 2025/5
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
94 replies
KevinYang2.71
Mar 21, 2025
SatisfiedMagma
Apr 7, 2025
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Isosceles Triangulation
worthawholebean   70
N Apr 2, 2025 by akliu
Source: USAMO 2008 Problem 4
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n - 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $ \mathcal{P}$ into $ n - 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
70 replies
worthawholebean
May 1, 2008
akliu
Apr 2, 2025
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Vertices of a pentagon invariant: 2011 USAMO #2
tenniskidperson3   51
N Apr 2, 2025 by akliu
An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
51 replies
tenniskidperson3
Apr 28, 2011
akliu
Apr 2, 2025
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The 2nd geometry problem is #20
Frestho   59
N Apr 1, 2025 by megahertz13
Source: 2020 AMC 10A #20 / 2020 AMC 12A #18
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?

$\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$
59 replies
Frestho
Jan 31, 2020
megahertz13
Apr 1, 2025
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Colored Pencils for Math Competitions
Owinner   17
N Apr 1, 2025 by lord_of_the_rook
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.
17 replies
Owinner
Mar 29, 2025
lord_of_the_rook
Apr 1, 2025
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USAJMO #5 - points on a circle
hrithikguy   207
N Mar 31, 2025 by LeYohan
Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P, A, C$ are collinear, and (iii) $DE \parallel AC$. Prove that $BE$ bisects $AC$.
207 replies
hrithikguy
Apr 28, 2011
LeYohan
Mar 31, 2025
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2025 INTEGIRLS NYC/NJ Math Competition
sargamsujit   3
N Mar 30, 2025 by Inaaya
NYC/NJ INTEGIRLS will be hosting our second annual math competition on May 3rd, 2025 from 9:30 AM to 4:30 PM EST at Rutgers University. Last year, we proudly organized the largest math competition for girls globally, welcoming over 500 participants from across the tristate area. Join other female-identifying and non-binary "STEMinists" in solving problems, socializing, playing games, and more! If you are interested in competing, please register at https://forms.gle/jqwEiq5PgqefetLj7

Find our website at https://nyc.nj.integirls.org/

[center]Important Information[/center]

Eligibility: This competition is open to all female-identifying and non-binary students in 8th grade or under. The competition is also completely free, including registration and lunch.

System: We will have two divisions: a middle school division and an elementary school division. There will be an individual round and team round. There will be prizes for the top competitors in each division!

Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The elementary school problems will range from introductory to AMC 8 level, while the middle school problems will be for more advanced problem-solvers. Team round problems will cover various difficulty levels.

Platform: This contest will be held in person at Rutgers University. Competitors will all receive free merchandise, raffle tickets, and the chance to win exclusive gift prizes!


[center]Prizes

Over $2,000 in awards, including plaques, medals, plushies, gift cards, toys, books, swag, and more for top competitors and teams

[center]Help Us Out[/center]


[center]Please help us in sharing our competition and spreading the word! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible and we would appreciate it if you could help us spread the word!
Format credits go to Indy INTEGIRLS!
3 replies
sargamsujit
Jan 28, 2025
Inaaya
Mar 30, 2025
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