Olympiad problems

2024 All-Russian Olympiad Regional Round, 9.2

Tags: quadratic , function graphing , conic , parabola , trapezoid , analytic geometry , geometry

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

2011 AIME Problems, 10

Tags: conic , parabola , probability

The probability that a set of three distinct vertices chosen at random from among the vertices of a regular $n$-gon determine an obtuse triangle is $\tfrac{93}{125}$. Find the sum of all possible values of $n$.

Kvant 2021, M2634

Tags: parabola , geometry

Consider a parabola. The [i]parabolic length[/i] of a segment is the length of the projection of this segment on a straight line perpendicular to the axis of symmetry of the parabola. In the parabola, two chords $AB$ and $CD$ are drawn, intersecting at the point $N{}$. Prove that the product of the parabolic lengths of the segments $AN$ and $BN$ is equal to the product of the parabolic lengths of the segments $CN$ and $DN$. [i]Proposed by M. Panov[/i]

1974 Putnam, A5

Tags: geometry , parabola , locus

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.

1981 Bulgaria National Olympiad, Problem 5

Tags: algebra , parabola , geometry

Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.

2021 Simon Marais Mathematical Competition, A1

Tags: conic , parabola , calculus

Let $a, b, c$ be real numbers such that $a \neq 0$. Consider the parabola with equation \[ y = ax^2 + bx + c, \] and the lines defined by the six equations \begin{align*} &y = ax + b, \quad & y = bx + c, \qquad \quad & y = cx + a, \\ &y = bx + a, \quad & y = cx + b, \qquad \quad & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$.

2007 Today's Calculation Of Integral, 192

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

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

2011 Today's Calculation Of Integral, 689

Tags: calculus , parabola , conic , geometry , rotation , geometric transformation , integration

Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis. Proposed by kunny

2019 Belarusian National Olympiad, 11.1

Tags: parabola , hyperbola , geometry , conic , circumcircle

[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points. [b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values. [i](I. Gorodnin)[/i]

2014 Belarusian National Olympiad, 1

Tags: geometry , parabola , conic

Let $ABC$ be a triangle inscribed in the parabola $y=x^2$ such that the line $AB \parallel$ the axis $Ox$. Also point $C$ is closer to the axis $Ox$ than the line $AB$. Given that the length of the segment $AB$ is 1 shorter than the length of the altitude $CH$ (of the triangle $ABC$). Determine the angle $\angle{ACB}$ .

2008 Purple Comet Problems, 7

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

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.

2004 Tournament Of Towns, 5

Tags: parabola , common tangent , circles , tangent , geometry

The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?

2004 AMC 12/AHSME, 18

Tags: parabola , conic

Points $ A$ and $ B$ are on the parabola $ y \equal{} 4x^2 \plus{} 7x \minus{} 1$, and the origin is the midpoint of $ \overline{AB}$. What is the length of $ \overline{AB}$? $ \textbf{(A)}\ 2\sqrt5 \qquad \textbf{(B)}\ 5\plus{}\frac{\sqrt2}{2} \qquad \textbf{(C)}\ 5\plus{}\sqrt2 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 5\sqrt2$

2022 JHMT HS, 6

Tags: parabola , conic , geometry

Triangle $ABC$ has side lengths $AC = 3$, $BC = 4$, and $AB = 5$. Let $I$ be the incenter of $\triangle{ABC}$, and let $\mathcal{P}$ be the parabola with focus $I$ and directrix $\overleftrightarrow{AC}$. Suppose that $\mathcal{P}$ intersects $\overline{AB}$ and $\overline{BC}$ at points $D$ and $E$, respectively. Find $DI+EI$.

2025 All-Russian Olympiad Regional Round, 10.10

Tags: parabola , algebra , geometry , function

On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals. [i]A. Tereshin[/i]

1984 AMC 12/AHSME, 22

Tags: conic , parabola , hyperbola

Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is A. a straight line B. a parabola C. part, but not all, of a parabola D. one branch of a hyperbola E. None of these

1963 AMC 12/AHSME, 29

Tags: parabola , conic

A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is: $\textbf{(A)}\ 800 \qquad \textbf{(B)}\ 640\qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 320 \qquad \textbf{(E)}\ 160$

2009 Today's Calculation Of Integral, 504

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

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.

2001 Putnam, 6

Tags: parabola , conic

Can an arc of a parabola inside a circle of radius $1$ have a length greater than $4$?

2012 Today's Calculation Of Integral, 771

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

(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis. (2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.

2019 India Regional Mathematical Olympiad, 6

Tags: coordinate geometry , parabola , conic , algebra , analytic geometry , geometry

Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.

2007 AMC 10, 10

Tags: geometry , conic , parabola

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}$

1990 IberoAmerican, 4

Tags: parabola , radical axis , conic , geometry , power of a point , rectangle

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$. a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies. b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.

1969 IMO Shortlist, 1

Tags: parabola , conic , locus , locus problems , geometry

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

2009 Today's Calculation Of Integral, 492

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

Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis. You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.