Olympiad problems

1954 Putnam, B4

Given the focus $F$ and the directrix $D$ of a parabola $P$ and a line $L$, describe a euclidean construction for the point or points of intersection of $P$ and $L.$ Be sure to identify the case for which there are no points of intersection.

2009 Today's Calculation Of Integral, 504

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

Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.

2003 AMC 12-AHSME, 25

Tags: algebra , function , domain , parabola , calculus , inequalities , conic

Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

2017 All-Russian Olympiad, 1

Tags: algebra , parabola , conic

$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ 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.

1987 Traian Lălescu, 1.1

Tags: parabola , geometry , conic

Consider the parabola $ P:x-y^2-(p+3)y-p=0,p\in\mathbb{R}^*. $ Show that $ P $ intersects the coordonate axis at three points, and that the circle formed by these three points passes through a fixed point.

2008 IberoAmerican Olympiad For University Students, 4

Tags: parabola , geometry , conic

Two vertices $A,B$ of a triangle $ABC$ are located on a parabola $y=ax^2 + bx + c$ with $a>0$ in such a way that the sides $AC,BC$ are tangent to the parabola. Let $m_c$ be the length of the median $CC_1$ of triangle $ABC$ and $S$ be the area of triangle $ABC$. Find \[\frac{S^2}{m_c^3}\]

2010 Contests, 2

Tags: inequalities , analytic geometry , parabola , algebra , polynomial , vieta , conic

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$

2004 239 Open Mathematical Olympiad, 8

Tags: geometry , circumcircle , parallelogram , parabola , ratio , geometric transformation , conic

Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$. [b]proposed by Sergej Berlov[/b]

2011 AMC 12/AHSME, 18

Tags: parabola , function , absolute value

Suppose that $|x+y|+|x-y|=2$. What is the maximum possible value of $x^2-6x+y^2$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9 $

2007 ITest, -1

Tags: analytic geometry , parabola , floor function , probability , number theory , conic , geometry

The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years). In order to receive credit for the correct answer to a problem, you must also correctly answer $\textit{every one}$ $\textit{of the previous parts}$ $\textit{of the Ultimate Question}$.

1996 Tournament Of Towns, (516) 3

Tags: conic , parabola

The parabola $y = x^2$ is drawn in the coordinate plane and then the axes are erased so that the whole parabola stays on the picture but the origin is not shown on it. Reconstruct the axes with compass and ruler alone. (A Egorov)

2014 AIME Problems, 6

Tags: quadratic , parabola , vieta , function , number theory , conic

The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.

1940 Putnam, A4

Tags: parabola , conic

Let $p$ be a real constant. The parabola $y^2=-4px$ rolls without slipping around the parabola $y^2=4px$. Find the equation of the locus of the vertex of the rolling parabola.

2005 IMC, 1

Tags: parabola , parameterization , conic

1. Let $f(x)=x^2+bx+c$, M = {x | |f(x)|<1}. Prove $|M|\leq 2\sqrt{2}$ (|...| = length of interval(s))

2005 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , parabola , conic

Two ants, one starting at $ (-1, 1) $, the other at $ (1, 1) $, walk to the right along the parabola $ y = x^2 $ such that their midpoint moves along the line $ y = 1 $ with constant speed $1$. When the left ant first hits the line $ y = \frac {1}{2} $, what is its speed?

2008 Purple Comet Problems, 7

Tags: analytic geometry , graphing lines , slope , parabola , algebra , polynomial , conic

A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

1996 VJIMC, Problem 1

Tags: parabola , geometry , conic

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

2011 Tournament of Towns, 4

Tags: parabola , regular polygon , combinatorial geometry , geometry , conic

Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for a) $N = 2011$ b) $N = 2012$ ?

2011 AIME Problems, 6

Tags: parabola , analytic geometry , geometry , geometric transformation , number theory , relatively prime , conic

Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Kyiv City MO 1984-93 - geometry, 1990.10.5

Tags: geometry , max , geometric inequality , analytic geometry , parabola , area

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of the quadrilateral $ABCD$.

2010 Purple Comet Problems, 26

Tags: analytic geometry , parabola , symmetry , graphing lines , slope , number theory , conic

In the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry of the parabola is a line with slope $\tfrac{r}{s}$ where r and s are relatively prime positive integers. Find $r + s$.

1958 AMC 12/AHSME, 22

Tags: parabola , analytic geometry , conic

A particle is placed on the parabola $ y \equal{} x^2 \minus{} x \minus{} 6$ at a point $ P$ whose $ y$-coordinate is $ 6$. It is allowed to roll along the parabola until it reaches the nearest point $ Q$ whose $ y$-coordinate is $ \minus{}6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $ x$-coordinates of $ P$ and $ Q$) is: $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$

2012 Today's Calculation Of Integral, 781

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

Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$. Note that the slope of $l$ is greater than that of $m$. (1) Exress the slope of $l$ in terms of $a$. (2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$. Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$. (3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.

2015 AMC 12/AHSME, 19

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? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2017 Flanders Math Olympiad, 1

Tags: analytic geometry , parabola , equal segments , area of triangle , conic

On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$