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1990 AMC 12 #24
dft   17
N Today at 2:34 AM by Bread10
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?
\[ \begin{tabular}{c c c c} {} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams and Baker} \\ \textbf{Boys:} & 71 & 81 & 79 \\ \textbf{Girls:} & 76 & 90 & ? \\ \textbf{Boys and Girls:} & 74 & 84 & \\ \end{tabular} \]
$ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 84 \quad\textbf{(E)}\ 85 $
17 replies
dft
Dec 31, 2011
Bread10
Today at 2:34 AM
J
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Metamorphosis of Medial and Contact Triangles
djmathman   101
N Mar 27, 2025 by ErTeeEs06
Source: 2014 USAJMO Problem 6
Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$.

(a) Prove that $I$ lies on ray $CV$.

(b) Prove that line $XI$ bisects $\overline{UV}$.
101 replies
djmathman
Apr 30, 2014
ErTeeEs06
Mar 27, 2025
J
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Bounded Quadratic
worthawholebean   38
N Mar 25, 2025 by SomeonecoolLovesMaths
Source: AIME 2010I Problem 6
Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3\]for all real numbers $ x$, and suppose $ P(11) = 181$. Find $ P(16)$.
38 replies
worthawholebean
Mar 17, 2010
SomeonecoolLovesMaths
Mar 25, 2025
J
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USAJMO problem 3: Inequality
BOGTRO   102
N Mar 21, 2025 by Marcus_Zhang
Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.
102 replies
BOGTRO
Apr 24, 2012
Marcus_Zhang
Mar 21, 2025
J
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Lots of ratios
austinchen2005   13
N Mar 6, 2025 by SomeonecoolLovesMaths
Source: 2020 AMC 10 B #3/2020 AMC 12 B #3
The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$?

$\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $
13 replies
austinchen2005
Feb 6, 2020
SomeonecoolLovesMaths
Mar 6, 2025
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maa REALLY loves rhombi
v4913   44
N Feb 21, 2025 by Magnetoninja
Source: 2023 AIME I #13
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
IMAGE
44 replies
v4913
Feb 8, 2023
Magnetoninja
Feb 21, 2025
J
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Numbers on a Blackboard
worthawholebean   64
N Feb 17, 2025 by HamstPan38825
Source: USAMO 2008 Problem 5
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
64 replies
worthawholebean
May 1, 2008
HamstPan38825
Feb 17, 2025
J
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The Miquel Point "Returns"
djmathman   110
N Feb 17, 2025 by xTimmyG
Source: 2013 USAJMO #3/USAMO #1
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.
110 replies
djmathman
Apr 30, 2013
xTimmyG
Feb 17, 2025
J
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Sines, Cosines, and Powers of 2... Oh My!
Lord.of.AMC   18
N Feb 2, 2025 by Shreyasharma
Source: 2013 AIME I Problem 14
For $\pi\leq\theta<2\pi$, let

\[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and
\[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
18 replies
Lord.of.AMC
Mar 15, 2013
Shreyasharma
Feb 2, 2025
J
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Congruent Incircles
worthawholebean   30
N Jan 21, 2025 by Mathandski
Source: AIME 2010I Problem 15
In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfrac{p}{q}$. Find $ p + q$.
30 replies
worthawholebean
Mar 17, 2010
Mathandski
Jan 21, 2025
J
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