Found problems: 487
The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.
The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil
attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?
Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{infinitely many}$
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
In a plane there is an ellipse $E$ with semiaxis $a,b$. Consider all the triangles inscribed in $E$ such that at least one of its sides is parallel to one of the axis of $E$.
Find both the locus of the centroid of all such triangles and its area.
Two branches of the hyperbola $xy=1$ are $C_1,C_2$ ($C_1$ in Quadrant I, $C_2$ in Quadrant III). Three apexes of regular triangle $PQR$ are on the hyperbola.
[b](a)[/b] $P,Q,R$ cannot be on the same branch.
[b](b)[/b] $P(-1,-1)$ is a point on $C_2$, if $Q,R$ are on $C_1$, find their coordinates.
For any real value of $ x$ the maximum value of $ 8x \minus{} 3x^2$ is:
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \frac83 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ \frac{16}{3}$
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
$P$ is a point on hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, if the distance from $P$ to right directrix is the arithmetic mean of the distance from $P$ to two focal points, then the $x$-axis of $P$ is________.
On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$.
[i]2011 Kyoto University entrance exam/Science, Problem 3[/i]
Solve for $z\in \mathbb{C}$ the equation :
\[ |z-|z+1||=|z+|z-1|| \]
Let $P_1,P_2,P_3,P_4$ be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that $P_1$ is tangent to $P_2$ at the point $q_2,P_2$ is tangent to $P_3$ at the point $q_3,P_3$ is tangent to $P_4$ at the point $q_4$, and $P_4$ is tangent to $P_1$ at the point $q_1$. Assume that all the points $q_1,q_2,q_3,q_4$ have distinct $x$-coordinates. Prove that $q_1,q_2,q_3,q_4$ lie on a graph of an at most quadratic polynomial.
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.
A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$.
[i]Evan O' Dorney.[/i]
A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.
An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.
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 $A_{n+j} = A_{j}$.)
Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane.
Note that the $x$ coodinate of $P$ is less than that of $Q$.
(a) If the origin $O$ is the midpoint of the lines egment $PQ$, then find the equation of the line $PQ$.
(b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$.
(c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.
Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?
For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$?
$ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$
Line $\frac{x}{4}+\frac{y}{3}=1$ and ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intersect at $A$ and $B$. A point on the ellipse $P$ satisties that the area of $\triangle PAB$ is $3$. The number of such points is
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$
A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:
$\textbf{(A)}\ 800 \qquad
\textbf{(B)}\ 640\qquad
\textbf{(C)}\ 400 \qquad
\textbf{(D)}\ 320 \qquad
\textbf{(E)}\ 160$
For the family of parabolas
$$y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a$$
(i) find the locus of vertices,
(ii) find the envelope,
(iii) sketch the envelope and two typical curves of the family.
Determine all monic polynomials $p(x)$ having real coefficients and satisfying the following two conditions:
$\bullet$ $p(x)$ is nonconstant, and all of its roots are distinct reals
$\bullet$ If $a $and $b$ are roots of $p(x)$ then $a + b + ab$ is also a root of $p(x)$.