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AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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force overlay inversion vibes
v4913   63
N Yesterday at 1:47 PM by starchan
Source: USAMO 2023/6
Let $ABC$ be a triangle with incenter $I$ and excenters $I_a$, $I_b$, and $I_c$ opposite $A$, $B$, and $C$, respectively. Let $D$ be an arbitrary point on the circumcircle of $\triangle{ABC}$ that does not lie on any of the lines $II_a$, $I_bI_c$, or $BC$. Suppose the circumcircles of $\triangle{DII_a}$ and $\triangle{DI_bI_c}$ intersect at two distinct points $D$ and $F$. If $E$ is the intersection of lines $DF$ and $BC$, prove that $\angle{BAD} = \angle{EAC}$.

Proposed by Zach Chroman
63 replies
v4913
Mar 23, 2023
starchan
Yesterday at 1:47 PM
Is EGMO good for JMO Geometry Questions?
MathRook7817   4
N Tuesday at 2:10 PM by MathRook7817
Hi guys, I was just wondering if EGMO is a good book for JMO/AMO/olympiad level questions, or if there exists another olympiad geo book. Thanks!
4 replies
MathRook7817
May 6, 2025
MathRook7817
Tuesday at 2:10 PM
Question about AMC 10
MathNerdRabbit103   15
N May 5, 2025 by GallopingUnicorn45
Hi,

Can anybody predict a good score that I can get on the AMC 10 this November by only being good at counting and probability, number theory, and algebra? I know some geometry because I took it in school though, but it isn’t competition math so it probably doesn’t count.

Thanks.
15 replies
MathNerdRabbit103
May 2, 2025
GallopingUnicorn45
May 5, 2025
Find the radius of circle O
TheMaskedMagician   3
N May 4, 2025 by fruitmonster97
Source: 1976 AHSME Problem 18
IMAGE

In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is

$\textbf{(A) }3+\sqrt{3}\qquad\textbf{(B) }15/\pi\qquad\textbf{(C) }9/2\qquad\textbf{(D) }2\sqrt{6}\qquad \textbf{(E) }\sqrt{22}$
3 replies
TheMaskedMagician
May 18, 2014
fruitmonster97
May 4, 2025
Tangent Circles in the Coordinate Plane
inventivedant   24
N May 4, 2025 by lpieleanu
Source: 2022 AMC 10B #22 / 2022 AMC 12B #21
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?

$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$
24 replies
inventivedant
Nov 17, 2022
lpieleanu
May 4, 2025
System
worthawholebean   10
N Apr 29, 2025 by daijobu
Source: AIME 2008II Problem 14
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m+n$.
10 replies
worthawholebean
Apr 3, 2008
daijobu
Apr 29, 2025
Angel Bisector and Equilateral Pentagon AMNPQ
El_Ectric   46
N Apr 29, 2025 by pikapika007
Source: 2016 USAMO 5
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$. Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}$. Denote by $\ell$ the angle bisector of $\angle MSQ$.

Prove that $\overline{OI}$ is parallel to $\ell$, where $O$ is the circumcenter of triangle $ABC$, and $I$ is the incenter of triangle $ABC$.
46 replies
El_Ectric
Apr 20, 2016
pikapika007
Apr 29, 2025
USAJMO #5 - points on a circle
hrithikguy   208
N Apr 28, 2025 by Adywastaken
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 $AC$.
208 replies
hrithikguy
Apr 28, 2011
Adywastaken
Apr 28, 2025
Complex numbers in geometry
v_Enhance   34
N Apr 28, 2025 by ESAOPS
Source: 2012 AIME I Problem 14
Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.
34 replies
v_Enhance
Mar 16, 2012
ESAOPS
Apr 28, 2025
Two Pentagons
r00tsOfUnity   54
N Apr 27, 2025 by ESAOPS
Source: 2023 AMC 10B #25 / 2023 AMC 12B #25
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$
54 replies
r00tsOfUnity
Nov 15, 2023
ESAOPS
Apr 27, 2025
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