Found problems: 487
2011 Today's Calculation Of Integral, 703
Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.
1957 AMC 12/AHSME, 43
We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $ x$-axis, the line $ x \equal{} 4$, and the parabola $ y \equal{} x^2$ is:
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 35\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 30\qquad
\textbf{(E)}\ \text{not finite}$
1969 Czech and Slovak Olympiad III A, 5
Two perpendicular lines $p,q$ and a point $A\notin p\cup q$ are given in plane. Find locus of all points $X$ such that \[XA=\sqrt{|Xp|\cdot|Xq|\,},\] where $|Xp|$ denotes the distance of $X$ from $p.$
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 CCA Math Bonanza, L3.4
Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$.
[i]2021 CCA Math Bonanza Lightning Round #3.4[/i]
2010 Kazakhstan National Olympiad, 1
Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle.
Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$
1973 Polish MO Finals, 6
Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.
1957 AMC 12/AHSME, 10
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\
\textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$
1976 Euclid, 4
Source: 1976 Euclid Part A Problem 4
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The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is
$\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$
2011 China Girls Math Olympiad, 3
The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.
2014 HMNT, 9
In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively.
Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.
1993 AMC 12/AHSME, 26
Find the largest positive value attained by the function
\[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \]
$ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $
2015 Balkan MO, 2
Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear.
[i](Cyprus)[/i]
2006 China Second Round Olympiad, 1
An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.
1969 IMO Longlists, 4
$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$
1962 AMC 12/AHSME, 26
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}$
2004 AMC 12/AHSME, 18
Points $ A$ and $ B$ are on the parabola $ y \equal{} 4x^2 \plus{} 7x \minus{} 1$, and the origin is the midpoint of $ \overline{AB}$. What is the length of $ \overline{AB}$?
$ \textbf{(A)}\ 2\sqrt5 \qquad
\textbf{(B)}\ 5\plus{}\frac{\sqrt2}{2} \qquad
\textbf{(C)}\ 5\plus{}\sqrt2 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 5\sqrt2$
2006 Iran Team Selection Test, 3
Let $l,m$ be two parallel lines in the plane.
Let $P$ be a fixed point between them.
Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$.
(By angle $EPF$ we mean the directed angle)
Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.
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 $ADE$.
PEN H Problems, 4
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.
Kvant 2024, M2823
A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane.
[i]P.A.Kozhevnikov[/i]
2006 Kyiv Mathematical Festival, 1
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Triangle $ABC$ and straight line $l$ are given at the plane. Construct using a compass and a ruler the straightline which is parallel to $l$ and bisects the area of triangle $ABC.$
2011 Olympic Revenge, 3
Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.
2013 Today's Calculation Of Integral, 870
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$
(1) Find all points of intersection of $E$ and $H$.
(2) Find the area of the region expressed by the system of inequality
\[\left\{
\begin{array}{ll}
3x^2+y^2\leq 3 &\quad \\
xy\geq \frac 34 , &\quad
\end{array}
\right.\]
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]