Found problems: 475
2011 ELMO Shortlist, 4
Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$.
[i]Calvin Deng.[/i]
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$.
2015 AMC 12/AHSME, 21
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?
$\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$
2001 Finnish National High School Mathematics Competition, 2
Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$
Prove that there is a line of the plane which does not meet either of the curves.
2010 USAJMO, 4
A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.
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?
1986 AMC 12/AHSME, 13
A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals
$ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$
2014 District Olympiad, 1
Solve for $z\in \mathbb{C}$ the equation :
\[ |z-|z+1||=|z+|z-1|| \]
2003 Mediterranean Mathematics Olympiad, 2
In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$
2013 Stanford Mathematics Tournament, 7
$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of parabola $y^2=x$. Compute the sum of all possible areas of $ABCD$.
2012 China Second Round Olympiad, 4
Let $F$ be the focus of parabola $y^2=2px(p>0)$, with directrix $l$ and two points $A,B$ on it. Knowing that $\angle AFB=\frac{\pi}{3}$, find the maximal value of $\frac{|MN|}{|AB|}$, where $M$ is the midpoint of $AB$ and $N$ is the projection of $M$ to $l$.
2009 Today's Calculation Of Integral, 494
Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant.
(1) Find the value of $ a$ and the coordinate of the point $ P$.
(2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$.
[color=green][Edited.][/color]
1979 Spain Mathematical Olympiad, 1
Calculate the area of the intersection of the interior of the ellipse $\frac{x^2}{16}+ \frac{y^2}{4}= 1$ with the circle bounded by the circumference $(x -2)^2 + (y - 1)^2 = 5$.
2004 239 Open Mathematical Olympiad, 8
Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$.
[b]proposed by Sergej Berlov[/b]
2013 Chile National Olympiad, 5
A conical surface $C$ is cut by a plane $T$ as shown in the figure on the back of this sheet. Show that $C \cap T$ is an ellipse. You can use as an aid the fact that if you consider the two spheres tangent to $C$ and $T$ as shown in the figure, they intersect $T$ in the bulbs.
[asy]
// calculate intersection of line and plane
// p = point on line
// d = direction of line
// q = point in plane
// n = normal to plane
triple lineintersectplan(triple p, triple d, triple q, triple n)
{
return (p + dot(n,q - p)/dot(n,d)*d);
}
// projection of point A onto line BC
triple projectionofpointontoline(triple A, triple B, triple C)
{
return lineintersectplan(B, B - C, A, B - C);
}
// calculate area of space triangle with vertices A, B, and C
real trianglearea(triple A, triple B, triple C)
{
return abs(cross(A - C, B - C)/2);
}
// calculate incentre of space triangle ABC
triple triangleincentre(triple A, triple B, triple C)
{
return (abs(B - C) * A + abs(C - A) * B + abs(A - B) * C)/(abs(B - C) + abs(C - A) + abs(A - B));
}
// calculate inradius of space triangle ABC
real triangleinradius(triple A, triple B, triple C)
{
return 2*trianglearea(A,B,C)/(abs(B - C) + abs(C - A) + abs(A - B));
}
// calculate excentre of space triangle ABC
triple triangleexcentre(triple A, triple B, triple C)
{
return (-abs(B - C) * A + abs(C - A) * B + abs(A - B) * C)/(-abs(B - C) + abs(C - A) + abs(A - B));
}
// calculate exradius of space triangle ABC
real triangleexradius(triple A, triple B, triple C)
{
return 2*trianglearea(A,B,C)/(-abs(B - C) + abs(C - A) + abs(A - B));
}
unitsize(2 cm);
pair project (triple A, real t) {
return((A.x, A.y*Sin(t) + A.z*Cos(t)));
}
real alpha, beta, theta, t;
real coneradius = 1, coneheight = 3;
real a, b, c;
real[] m, r;
triple A, B, V;
triple ellipsecenter, ellipsex, ellipsey;
triple[] F, O, P, R, W;
path[] ellipse, spherering;
theta = 15;
V = (0,0,-coneheight);
m[1] = sqrt(Cos(theta)^2*coneheight^2 - Sin(theta)^2*coneradius^2)/coneradius;
m[2] = -m[1];
alpha = -aTan(Sin(theta)/m[1]);
beta = -aTan(Sin(theta)/m[2]) + 180;
A = (coneradius*Cos(alpha), coneradius*Sin(alpha), 0);
B = (coneradius*Cos(beta), coneradius*Sin(beta), 0);
W[1] = interp(V,(coneradius,0,0),0.6);
W[2] = interp(V,(-coneradius,0,0),0.4);
O[1] = triangleexcentre(V,W[1],W[2]);
O[2] = triangleincentre(V,W[1],W[2]);
r[1] = triangleexradius(V,W[1],W[2]);
r[2] = triangleinradius(V,W[1],W[2]);
F[1] = projectionofpointontoline(O[1],W[1],W[2]);
F[2] = projectionofpointontoline(O[2],W[1],W[2]);
P[1] = O[1] - (0,0,r[1]*coneradius/sqrt(coneradius^2 + coneheight^2));
P[2] = O[2] - (0,0,r[2]*coneradius/sqrt(coneradius^2 + coneheight^2));
spherering[11] = shift(project(P[1],theta))*yscale(Sin(theta))*arc((0,0),r[1]*coneheight/sqrt(coneradius^2 + coneheight^2),alpha,beta);
spherering[12] = shift(project(P[1],theta))*yscale(Sin(theta))*arc((0,0),r[1]*coneheight/sqrt(coneradius^2 + coneheight^2),beta,alpha + 360);
spherering[21] = shift(project(P[2],theta))*yscale(Sin(theta))*arc((0,0),r[2]*coneheight/sqrt(coneradius^2 + coneheight^2),alpha,beta);
spherering[22] = shift(project(P[2],theta))*yscale(Sin(theta))*arc((0,0),r[2]*coneheight/sqrt(coneradius^2 + coneheight^2),beta,alpha + 360);
ellipsecenter = (W[1] + W[2])/2;
a = abs(W[1] - ellipsecenter);
c = abs(F[1] - ellipsecenter);
b = sqrt(a^2 - c^2);
ellipsex = (W[1] - W[2])/abs(W[1] - W[2]);
ellipsey = (0,1,0);
ellipse[1] = project(ellipsecenter + a*ellipsex, theta);
for (t = 0; t <= 180; t = t + 5) {
ellipse[1] = ellipse[1]--project(ellipsecenter + a*Cos(t)*ellipsex + b*Sin(t)*ellipsey, theta);
}
ellipse[2] = project(ellipsecenter - a*ellipsex, theta);
for (t = 180; t <= 360; t = t + 5) {
ellipse[2] = ellipse[2]--project(ellipsecenter + a*Cos(t)*ellipsex + b*Sin(t)*ellipsey, theta);
}
R[1] = ellipsecenter + 1*ellipsex + ellipsey;
R[2] = ellipsecenter - 1.2*ellipsex + ellipsey;
R[3] = ellipsecenter - 1*ellipsex - ellipsey;
R[4] = ellipsecenter + 1.2*ellipsex - ellipsey;
fill(ellipse[1]--ellipse[2]--cycle, gray(0.9));
draw(yscale(Sin(theta))*Circle((0,0),coneradius));
draw(project(V,theta)--project(A,theta));
draw(project(V,theta)--project(B,theta));
draw(Circle(project(O[1],theta),r[1]));
draw(Circle(project(O[2],theta),r[2]));
draw(spherering[11], dashed);
draw(spherering[12]);
draw(spherering[21], dashed);
draw(spherering[22]);
draw(ellipse[1], dashed);
draw(ellipse[2]);
draw(project(R[1],theta)--interp(project(R[1],theta),project(R[2],theta),0.13));
draw(interp(project(R[1],theta),project(R[2],theta),0.13)--interp(project(R[1],theta),project(R[2],theta),0.76), dashed);
draw(interp(project(R[1],theta),project(R[2],theta),0.76)--project(R[2],theta));
draw(project(R[2],theta)--project(R[3],theta)--project(R[4],theta)--project(R[1],theta));
label("$C$", (-1,0.3));
label("$T$", (1.2,-0.8));
dot(project(F[1],theta));
dot(project(F[2],theta));
//dot("$F_1$", project(F[1],theta));
//dot("$F_2$", project(F[2],theta));
//dot("$O_1$", project(O[1],theta));
//dot("$O_2$", project(O[2],theta));
//dot("$P_1$", project(P[1],theta));
//dot("$V$", project(V,theta));
//dot("$W_1$", project(W[1],theta));
//dot("$W_2$", project(W[2],theta));
[/asy]
1952 Putnam, B6
Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.
1992 National High School Mathematics League, 1
For any positive integer $n$, $A_n$ and $B_n$ are intersection of parabola $y=(n^2+n)x^2-(2n+1)x+1$ and $x$-axis. Then, the value of $|A_1B_1|+|A_2B_2|+\cdots+|A_{1992}B_{1992}|$ is
$\text{(A)}\frac{1991}{1992}\qquad\text{(B)}\frac{1992}{1993}\qquad\text{(C)}\frac{1991}{1993}\qquad\text{(D)}\frac{1993}{1992}$
2018 Caucasus Mathematical Olympiad, 6
Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.
1979 IMO Shortlist, 22
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$
2007 ITest, 51
Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]
2014 AIME Problems, 6
The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
1998 USAMO, 6
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)
2012 Today's Calculation Of Integral, 790
Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$.
(1) Find the equations of the tangents $l_1,\ l_2$.
(2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.
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?
1982 All Soviet Union Mathematical Olympiad, 339
There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?