Olympiad problems

Today's calculation of integrals, 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 $\frac{11}{3}.$

1996 VJIMC, Problem 1

Tags: parabola , geometry , conic

Is it possible to cover the plane with the interiors of a finite number of parabolas?

2025 Belarusian National Olympiad, 8.7

Tags: parabola , algebra

Yan and Kirill play a game. At first Kirill says 4 numbers $x_1<x_2<x_3<x_4$, and then Yan says three pairwise different non zero numbers $a_1$, $a_2$ and $a_3$. For all $i$ from $1$ to $3$ they consider the quadratic trinomial $f_i(x)$ which has roots $x_i$ and $x_{i+1}$ and leading coefficient $a_i$, and construct on the plane the graphs of that trinomials. Yan wins if in every pair $(f_1(x),f_2(x))$ and $(f_2(x),f_3(x))$ their graphs intersect at exactly one point, and if in some pair graphs do not intersect or intersect at more than one point Kirill wins. Find which player can guarantee his win regardless of the actions of his opponent. [i]V. Kamianetski[/i]

2009 Stanford Mathematics Tournament, 9

Tags: parabola , calculus , algebra , polynomial , derivative , quadratic , conic

Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$

2012 Iran MO (3rd Round), 5

Tags: parabola , geometry , conic

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]

2010 Sharygin Geometry Olympiad, 22

Tags: parabola , geometry , geometric transformation , reflection , trapezoid , symmetry , conic

A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.

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]

2015 AMC 12/AHSME, 12

Tags: parabola , analytic geometry , geometry , conic

The parabolas $y=ax^2-2$ and $y=4-bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$? $\textbf{(A) }1\qquad\textbf{(B) }1.5\qquad\textbf{(C) }2\qquad\textbf{(D) }2.5\qquad\textbf{(E) }3$

2016 AMC 10, 9

Tags: parabola , conic

All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

1986 AMC 12/AHSME, 13

Tags: quadratic , parabola , conic

A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$

1957 AMC 12/AHSME, 40

Tags: parabola , irrational number , conic

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}$

2007 Today's Calculation Of Integral, 192

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

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 $\frac{S(t)}{\sqrt{t}}.$

2007 AMC 10, 10

Tags: geometry , parabola , conic

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}$

2013 ISI Entrance Examination, 8

Tags: parabola , analytic geometry , conic

Let $ABCD$ be a square such that $AB$ lies along the line $y=x+8,$ and $C$ and $D$ lie on the parabola $y=x^2.$ Find all possible values of sidelength of the square.

1949 Putnam, A1

Tags: 3d geometry , parabola , surfaces

Answer either (i) or (ii): (i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line. (II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?

2013 AMC 12/AHSME, 21

Tags: parabola , geometry , angle bisector , conic , combinatorics

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 $

1974 Putnam, A5

Tags: locus , parabola , geometry

Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.

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 $\frac{11}{3}.$

2006 AMC 12/AHSME, 12

Tags: parabola , analytic geometry , calculus , algebra , system of equations , conic

The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$? $ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$

2013 AMC 12/AHSME, 20

Tags: trigonometry , geometry , trapezoid , parabola , analytic geometry , graphing lines , conic

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

2013 Hitotsubashi University Entrance Examination, 3

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

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

1962 AMC 12/AHSME, 19