Found problems: 253
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular $n$-gon determine an obtuse triangle is $\tfrac{93}{125}$. Find the sum of all possible values of $n$.
$ABCD$ 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 $ABCD$.
Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.
Given triangle $ ABC$ with base $ AB$ fixed in length and position. As the vertex $ C$ moves on a straight line, the intersection point of the three medians moves on:
$ \textbf{(A)}\ \text{a circle} \qquad
\textbf{(B)}\ \text{a parabola} \qquad
\textbf{(C)}\ \text{an ellipse} \qquad
\textbf{(D)}\ \text{a straight line} \qquad
\textbf{(E)}\ \text{a curve here not listed}$
[asy]
size(180);
defaultpen(linewidth(0.8));
real r=4/5;
draw((-1,0)..(-6/7,r/3)..(0,r)..(6/7,r/3)..(1,0),linetype("4 4"));
draw((-1,0)--(1,0)^^origin--(0,r));
label("$A$",(-1,0),W);
label("$B$",(1,0),E);
label("$M$",origin,S);
label("$C$",(0,r),N);
[/asy]
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is:
$\textbf{(A) }1\qquad
\textbf{(B) }15\qquad
\textbf{(C) }15\tfrac13\qquad
\textbf{(D) }15\tfrac12\qquad
\textbf{(E) }15\tfrac34$
Various points $x_1,..., x_n$ ($n \ge 3$) are randomly located on the $Ox$ axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let$ y = f_1$, $...$ , $y = f_m$ are functions that define these parabolas. Prove that the parabola $y = f_1 +...+ f_m$ intersects the $Ox$ axis at two points.
$A(0,2)$, and two points $B,C$ on parabola $y^2=x+4$ satisfy that $AB\perp BC$. Find the range value of $y_C$.
Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which
\[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1.
\]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$?
$ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$
The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is:
$ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$
On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equall likely to be hit, find the probability that hte point hit is nearer to the center than any edge.
In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant.
(1) Find the equation of $ l$.
(2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.
Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$
Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$,
[b](a)[/b] $|f(x)| \leq 5/4$,
[b](b)[/b] $|g(x)| \leq 2$.
Answer either (i) or (ii):
(i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line.
(II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?
Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.
Let $P_1$, $P_2$, $P_3$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $P_i$. In other words, find the maximum number of points that can lie on two or more of the parabolas $P_1$, $P_2$, $P_3$ .
The graph $y=x^2+ax+b$ intersects any of the two axes at points $A$, $B$, and $C$. The incenter of triangle $ABC$ lies on the line $y=x$. Prove that $a+b+1=0$.
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?
Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).
In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$.
[i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]