Found problems: 475
1984 Putnam, B2
Find the minimum value of\[ (u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2 \]for $0<u<\sqrt{2}$ and $v>0$
1982 IMO Longlists, 56
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] $|g(x)| \leq 2$.
2011 Math Prize For Girls Problems, 16
Let $N$ be the number of ordered pairs of integers $(x, y)$ such that
\[
4x^2 + 9y^2 \le 1000000000.
\]
Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$ ?
2016 Israel Team Selection Test, 3
Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).
1985 IMO Longlists, 31
Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.
Today's calculation of integrals, 862
Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$
2007 Today's Calculation Of Integral, 193
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.
2009 Today's Calculation Of Integral, 493
In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse.
(1) Find the equation of $ l$.
(2) Express $ S$ in terms of $ a,\ b$.
(3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.
1999 National High School Mathematics League, 6
Points $A(1,2)$, a line that passes $(5,-2)$ intersects the parabola $y^2=4x$ at two points $B,C$. Then, $\triangle ABC$ is
$\text{(A)}$ an acute triangle
$\text{(B)}$ an obtuse triangle
$\text{(C)}$ a right triangle
$\text{(D)}$ not sure
1997 National High School Mathematics League, 8
Line $l$ that passes right focal point of hyperbola $x^2-\frac{y^2}{2}=1$ intersects the hyperbola at $A,B$. The number of line $l$ that $|AB|=\lambda$ is 3, then $\lambda=$________.
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 $p.$
1942 Putnam, A4
Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$. At what angle do the curves of one family cut the curves of the other family at the origin?
2007 Harvard-MIT Mathematics Tournament, 4
Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.
2009 Purple Comet Problems, 22
The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.
[asy]
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real f(real x) {return 1.2*exp(2/3*log(16-x^2));}
path Q=graph(f,-3.99999,3.99999);
path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle};
for(int k=0;k<3;++k)
{
fill(P[k],grey); draw(P[k]);
}
draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]
1985 AIME Problems, 11
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
1999 Spain Mathematical Olympiad, 1
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 $m$.
2023 Indonesia TST, 3
Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.
2005 China National Olympiad, 2
A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.
2007 Today's Calculation Of Integral, 211
When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves,
prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.
2017 All-Russian Olympiad, 1
$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals.
2009 Today's Calculation Of Integral, 476
Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.
2021 China Second Round Olympiad, Problem 12
Let $C$ be the left vertex of the ellipse $\frac{x^2}8+\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$.
(i) Compute the maximum of $|CA|+|CB|$.
(ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and $N$, respectively. If the intersection of the perpendicular bisector of $MN$ and the circle with diameter $MN$ lies inside the given ellipse, compute the range of possible values of $k$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 12)[/i]
2005 Czech-Polish-Slovak Match, 5
Given a convex quadrilateral $ABCD$, find the locus of the points $P$ inside the quadrilateral such that
\[S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}\]
(where $S_X$ denotes the area of triangle $X$).
2019 China National Olympiad, 4
Given an ellipse that is not a circle.
(1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area is unique.
(2) Construct this rhombus using a compass and a straight edge.
2024 ISI Entrance UGB, P7
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.