Found problems: 475
2011 Today's Calculation Of Integral, 731
Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively.
When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.
2000 Croatia National Olympiad, Problem 1
Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.
1957 AMC 12/AHSME, 40
If the parabola $ y \equal{} \minus{}x^2 \plus{} bx \minus{} 8$ has its vertex on the $ x$-axis, then $ b$ must be:
$ \textbf{(A)}\ \text{a positive integer}\qquad \\
\textbf{(B)}\ \text{a positive or a negative rational number}\qquad \\
\textbf{(C)}\ \text{a positive rational number}\qquad \\
\textbf{(D)}\ \text{a positive or a negative irrational number}\qquad \\
\textbf{(E)}\ \text{a negative irrational number}$
1955 AMC 12/AHSME, 39
If $ y\equal{}x^2\plus{}px\plus{}q$, then if the least possible value of $ y$ is zero $ q$ is equal to:
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \frac{p^2}{4} \qquad
\textbf{(C)}\ \frac{p}{2} \qquad
\textbf{(D)}\ \minus{}\frac{p}{2} \qquad
\textbf{(E)}\ \frac{p^2}{4}\minus{}q$
1942 Putnam, B2
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.
2005 AMC 12/AHSME, 8
For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$
2021 Simon Marais Mathematical Competition, A1
Let $a, b, c$ be real numbers such that $a \neq 0$. Consider the parabola with equation
\[ y = ax^2 + bx + c, \]
and the lines defined by the six equations
\begin{align*}
&y = ax + b, \quad & y = bx + c, \qquad \quad & y = cx + a, \\
&y = bx + a, \quad & y = cx + b, \qquad \quad & y = ax + c.
\end{align*}
Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$.
2009 International Zhautykov Olympiad, 1
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.
2007 AMC 10, 10
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}$
1994 Polish MO Finals, 2
Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.
2013 AMC 12/AHSME, 21
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 $
2007 Tournament Of Towns, 1
$A,B,C$ and $D$ are points on the parabola $y = x^2$ such that $AB$ and $CD$ intersect on the $y$-axis. Determine the $x$-coordinate of $D$ in terms of the $x$-coordinates of $A,B$ and $C$, which are $a, b$ and $c$ respectively.
2007 Today's Calculation Of Integral, 177
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 $\frac{S_{1}}{S_{2}}.$
2020 Belarusian National Olympiad, 11.3
Four points $A$, $B$, $C$, $D$ lie on the hyperbola $y=\frac{1}{x}$. In triangle $BCD$ the point $A_1$ is the circumcenter of the triangle, which vertices are the midpoints of sides of $BCD$. In triangles $ACD$, $ABD$ and $ABC$ points $B_1$, $C_1$ and $D_1$ are chosen similarly. It turned out that points $A_1$, $B_1$, $C_1$ and $D_1$ are pairwise different and concyclic.
Prove that the center of that circle coincides with the $(0,0)$ point.
2014 VJIMC, Problem 4
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.
2009 Today's Calculation Of Integral, 419
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$.
2012 AMC 12/AHSME, 10
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$?
${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $
2011 China Second Round Olympiad, 7
The line $x-2y-1=0$ insects the parabola $y^2=4x$ at two different points $A, B$. Let $C$ be a point on the parabola such that $\angle ACB=\frac{\pi}{2}$. Find the coordinate of point $C$.
Revenge EL(S)MO 2024, 5
In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic.
Proposed by [i]Alex Wang[/i]
1984 AMC 12/AHSME, 22
Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is
A. a straight line
B. a parabola
C. part, but not all, of a parabola
D. one branch of a hyperbola
E. None of these
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]
2012 Today's Calculation Of Integral, 781
Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$.
Note that the slope of $l$ is greater than that of $m$.
(1) Exress the slope of $l$ in terms of $a$.
(2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$.
Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$.
(3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.
2000 Harvard-MIT Mathematics Tournament, 7
Let $ABC$ be a triangle inscribed in the ellipse $\frac{x^2}{4} +\frac{y^2}{9}= 1$. If its centroid is the origin $(0,0)$, find its area.
2019 AMC 12/AHSME, 6
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?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$
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 $ADE$.