Found problems: 137
2011 Math Prize For Girls Problems, 20
Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\dfrac{1}{\sqrt[4]{3}}\,$. What is the value of $900p$, rounded to the nearest integer?
1988 IberoAmerican, 3
Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.
1950 AMC 12/AHSME, 49
A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $ 1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is:
$\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\
\textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\
\textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\
\textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\
\textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$
2005 Taiwan TST Round 2, 3
In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.
1999 National High School Mathematics League, 14
Given $A(-2,2)$, and $B$ is a moving point on ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$. $F$ is the left focal point of the ellipse, find the coordinate of $B$ when $|AB|+\frac{5}{3}|BF|$ takes its minumum value.
2012 ELMO Problems, 5
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
2012 Math Prize For Girls Problems, 16
Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?
2006 China Second Round Olympiad, 1
An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.
1994 Putnam, 2
Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{x}{2}$, the x-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx,$ the y-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1.$
2008 AMC 10, 19
A cylindrical tank with radius $ 4$ feet and height $ 9$ feet is lying on its side. The tank is filled with water to a depth of $ 2$ feet. What is the volume of the water, in cubic feet?
$ \textbf{(A)}\ 24\pi \minus{} 36 \sqrt {2} \qquad \textbf{(B)}\ 24\pi \minus{} 24 \sqrt {3} \qquad \textbf{(C)}\ 36\pi \minus{} 36 \sqrt {3} \qquad \textbf{(D)}\ 36\pi \minus{} 24 \sqrt {2} \\ \textbf{(E)}\ 48\pi \minus{} 36 \sqrt {3}$
2008 AIME Problems, 13
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
2023 ELMO Shortlist, G7
Let \(\mathcal E\) be an ellipse with foci \(F_1\) and \(F_2\), and let \(P\) be a point on \(\mathcal E\). Suppose lines \(PF_1\) and \(PF_2\) intersect \(\mathcal E\) again at distinct points \(A\) and \(B\), and the tangents to \(\mathcal E\) at \(A\) and \(B\) intersect at point \(Q\). Show that the midpoint of \(\overline{PQ}\) lies on the circumcircle of \(\triangle PF_1F_2\).
[i]Proposed by Karthik Vedula[/i]
1986 National High School Mathematics League, 7
Inside a circular column with a bottom surface with radius of $6$, there are two balls with radius of $6$ as well. The distance betwen their centers is $13$. Draw a plane that is tangent to both spherical surface, intersect the circular column at a curve $C$. $C$ is a ellipse, then the sum of its short axis and long axis is________.
2001 National High School Mathematics League, 7
The length of minor axis of ellipse $\rho-\frac{1}{2-\cos\theta}$ is________.
2000 China Team Selection Test, 1
Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.
2012 ELMO Shortlist, 5
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
2003 Polish MO Finals, 5
The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.
1985 ITAMO, 11
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
2012 Canada National Olympiad, 3
Let $ABCD$ be a convex quadrilateral and let $P$ be the point of intersection of $AC$ and $BD$. Suppose that $AC+AD=BC+BD$. Prove that the internal angle bisectors of $\angle ACB$, $\angle ADB$ and $\angle APB$ meet at a common point.
2023 Sharygin Geometry Olympiad, 23
An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.
2019 AMC 10, 10
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$
1964 AMC 12/AHSME, 2
The graph of $x^2-4y^2=0$ is:
${{ \textbf{(A)}\ \text{a parabola} \qquad\textbf{(B)}\ \text{an ellipse} \qquad\textbf{(C)}\ \text{a pair of straight lines} \qquad\textbf{(D)}\ \text{a point} }\qquad\textbf{(E)}\ \text{none of these} } $
2007 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ an isosceles triangle, $P$ a point belonging to its interior. Denote $M$, $N$ the intersection points of the circle $\mathcal{C}(A, AP)$ with the sides $AB$ and $AC$, respectively.
Find the position of $P$ if $MN+BP+CP$ is minimum.
1941 Putnam, B4
Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$. Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$.
Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.
2017 BMT Spring, 19
Let $T$ be the triangle in the $xy$-plane with vertices $(0, 0)$, $(3, 0)$, and $\left(0, \frac32\right)$. Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$.