Found problems: 487
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.
Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$. Find $w_1+w_2$.
What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?
Let there be a triangle $\triangle ABC$ with $BC=a$, $CA=b$, $AB=c$.
Let $T$ be a point not inside $\triangle ABC$ and on the same side of $A$ with respect to $BC$, such that $BT-CT=c-b$.
Let $n=BT$ and $m=CT$. Find the point $P$ that minimizes $f(P)=-a \cdot AP + m \cdot BP + n \cdot CP$.
For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality.
\[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]
The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
[i]Proposed by Sammy Luo[/i]
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 $\frac{11}{3}.$
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?
$\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$
Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.
The graph of $ 2x^2 \plus{} xy \plus{} 3y^2 \minus{} 11x \minus{} 20y \plus{} 40 \equal{} 0$ is an ellipse in the first quadrant of the $ xy$-plane. Let $ a$ and $ b$ be the maximum and minimum values of $ \frac {y}{x}$ over all points $ (x, y)$ on the ellipse. What is the value of $ a \plus{} b$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ \frac72 \qquad \textbf{(D)}\ \frac92 \qquad \textbf{(E)}\ 2\sqrt {14}$
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$.
[b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.
[b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$
There is a right angle whose vertex moves on a fixed circle and one of it's sides passes a fixed point. What is the curve that the other side of the angle is always tangent to it.
The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD.
Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$.
Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$
In a plane:
1. An ellipse with foci $F_1$, $F_2$ lies inside a circle $\omega$. Construct a chord $AB$ of $\omega$. touching the ellipse and such that $A$, $B$, $F_1$, and $F_2$ are concyclic.
2. Let a point $P$ lie inside an acute angled triangle $ABC$, and $A'$, $B'$, $C'$ be the projections of $P$ to $BC$, $CA$, $AB$ respectively. Prove that the diameter of circle $A'B'C'$ equals $CP$ if and only if the circle $ABP$ passes through the circumcenter of $ABC$.
[i]Proposed by Alexey Zaslavsky[/i]
[img]https://cdn.artofproblemsolving.com/attachments/8/e/ac4a006967fb7013efbabf03e55a194cbaa18b.png[/img]
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer?
[asy]
import graph;
size(300);
defaultpen(linewidth(0.8)+fontsize(10));
real k=1.5;
real endp=sqrt(k);
real f(real x) {
return k-x^2;
}
path parabola=graph(f,-endp,endp)--cycle;
filldraw(parabola, lightgray);
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));
label("Region I", (0,2*k/5));
label("Box II", (51/64*endp,13/16*k));
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));
[/asy]
Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space $ \mathbb{E}^3$ not lying on the $ z$-axis by the metric tensor \[ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \minus{}\sqrt{x^2\plus{}y^2} \\ \end{array} \right),\] where $ (x,y,z)$ is a Cartesian coordinate system $ \mathbb{E}^3$. Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the $ (x,y)$-plane are straight lines or conic sections with focus at the origin
[i]P. Nagy[/i]
A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree?
[asy]
filldraw(circle((0,0),1),gray(0.7));
draw((0,0)--(0.81915, 0.57358));
dot((0.81915, 0.57358));
draw((0.81915, 0.57358)--(0.475006, 1.06507));
arrow((0.417649,1.14698), dir(305), 12);
draw((0,0)--(1,0),dashed);
label("$\theta$", (0.2, 0.2/3), fontsize(8));
label("$r$", (0.409575,0.28679), NW, fontsize(8));
[/asy]
[i]Problem proposed by Ahaan Rungta[/i]
Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$. A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$. $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$.
Prove that \[ FB_1 + FB_2 + \ldots + FB_n > np . \]
[i]Calin Popescu[/i]
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.
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 $\frac{11}{3}.$
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.
How many normals to an ellipse can be drawn from a given point in plane? Find the region in which the number of normals is maximal.