Olympiad problems

2007 Today's Calculation Of Integral, 177

Tags: calculus , integration , parabola , geometry , vector , conic

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 Tuymaada Olympiad, 5

Tags: parabola , construction , geometry , algebra , analytic geometry , quadratic , conic

Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?

1985 AIME Problems, 11

Tags: ellipse , analytic geometry , geometry , geometric transformation , reflection , conic

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?

2015 AMC 10, 24

Tags: geometry , perimeter , calculus , rhombus , parabola , conic

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2013 Princeton University Math Competition, 5

Tags: parabola , conic

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 $w_1+w_2$.

2009 Today's Calculation Of Integral, 470

Tags: calculus , integration , geometry , parabola , calculus computations , conic

Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.

1992 Flanders Math Olympiad, 3

Tags: geometry , 3d geometry , sphere , conic

a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?

2005 Iran MO (3rd Round), 5

Tags: circumcircle , euler , projective geometry , geometry , conic

Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.

1998 All-Russian Olympiad, 5

Tags: parabola , quadratic , algebra , geometry , conic

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 $\omega_{1998}$.

2008 AMC 12/AHSME, 17

Tags: parabola , geometry , analytic geometry , calculus , graphing lines , slope , conic

Let $ A$, $ B$, and $ C$ be three distinct points on the graph of $ y\equal{}x^2$ such that line $ AB$ is parallel to the $ x$-axis and $ \triangle{ABC}$ is a right triangle with area $ 2008$. What is the sum of the digits of the $ y$-coordinate of $ C$? $ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 20$

2013 Today's Calculation Of Integral, 891

Tags: calculus , integration , geometry , 3d geometry , hyperbola , rotation , conic

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

2005 South East Mathematical Olympiad, 1

Tags: parabola , function , calculus , derivative , algebra , conic

Let $a \in \mathbb{R}$ be a parameter. (1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola. (2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

1986 Balkan MO, 4

Tags: geometry , perimeter , hyperbola , parallelogram , rectangle , perpendicular bisector , conic

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

2019 AMC 12/AHSME, 6

Tags: ellipse , conic

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? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

2006 Stanford Mathematics Tournament, 13

Tags: trigonometry , hyperbola , analytic geometry , graphing lines , slope , conic

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

2000 Croatia National Olympiad, Problem 1

Tags: conic section , geometry , parabola , conic

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 $T'$.

2013 Today's Calculation Of Integral, 862

Tags: calculus , integration , analytic geometry , graphing lines , slope , parabola , conic

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 $\frac{11}{3}.$

2008 Harvard-MIT Mathematics Tournament, 3

Tags: quadratic , inequalities , function , parabola , analytic geometry , absolute value , conic

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2003 Polish MO Finals, 5

Tags: geometry , 3d geometry , sphere , tetrahedron , circumcircle , ellipse , conic

The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.

1991 Arnold's Trivium, 7

Tags: ellipse , conic

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.

2010 ELMO Shortlist, 3

Tags: incenter , circumcircle , parabola , geometry , conic

A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$. [i]Evan O' Dorney.[/i]

Russian TST 2017, P1

Tags: trapezoid , circumcircle , moving points , conic , projective geometry , geometry

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

1999 Baltic Way, 5

Tags: parabola , analytic geometry , slope , quadratic , algebra , conic

The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.

2013 Princeton University Math Competition, 3

Tags: geometry , parabola , analytic geometry , graphing lines , slope , calculus , conic

The area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-25$, can be expressed as $\tfrac ab\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?

2012 Iran MO (3rd Round), 5

Tags: parabola , conic , geometry

Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$. [i]Proposed by Nima Hamidi[/i]