Found problems: 253
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.
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]
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.
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 $PQ$.
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 $\frac{S_{1}}{S_{2}}.$
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.
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
${ \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D}}\ 840\qquad\textbf{(E)}\ 870 $
The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when:
$ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$
The parabola $y = x^2$ is drawn in the coordinate plane and then the axes are erased so that the whole parabola stays on the picture but the origin is not shown on it. Reconstruct the axes with compass and ruler alone.
(A Egorov)
Two distinct points $A$ and $B$ are marked on the left half of the parabola $y = x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M, N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM - KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above.
(I. Voronovich)
Two points $ B$ and $ C$ are in a plane. Let $ S$ be the set of all points $ A$ in the plane for which $ \triangle ABC$ has area $ 1$. Which of the following describes $ S$?
$ \textbf{(A)}\ \text{two parallel lines}\qquad
\textbf{(B)}\ \text{a parabola}\qquad
\textbf{(C)}\ \text{a circle}\qquad
\textbf{(D)}\ \text{a line segment}\qquad
\textbf{(E)}\ \text{two points}$
A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.
All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$?
$ \textbf{(A)}\ 14\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 17\qquad
\textbf{(E)}\ 18$
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?
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 $l$.
Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for
a) $N = 2011$
b) $N = 2012$ ?
Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.
Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value:
$S = |n-1| + |n-2| + \ldots + |n-100|$
Let $ ABC$ be an acute triangle. Point $ D$ lies on side $ BC$. Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively. Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$. Let $ H$ be the orthocenter of triangle $ ABC$. Prove that $ \angle{DAX} \equal{} \angle{DAH}$.
[i]Zuming Feng.[/i]
Suppose that $|x+y|+|x-y|=2$. What is the maximum possible value of $x^2-6x+y^2$?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9
$
Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$
$ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $