Olympiad problems

2011 Albania Team Selection Test, 1

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.

2005 National High School Mathematics League, 15

Tags: parabola , analytic geometry , geometry , conic

$A(1,1)$ is a point on parabola $y=x^2$. Draw the tangent line of the parabola that passes $A$, the line intersects $x$-axis at $D$, intersects $y$-axis at $B$. $C$ is a point on the parabola, and $E$ is a point on segment $AC$, such that $\frac{AE}{EC}=\lambda_1$, $F$ is a point on segment $BC$, such that $\frac{BF}{FC}=\lambda_2$. If $\lambda_1+\lambda_2=1$, $CD$ and $EF$ intersect at $P$. When $C$ moves, find the path equation of $P$.

1959 AMC 12/AHSME, 8

Tags: quadratic , parabola , analytic geometry , optimizing quadratics , algebra , conic

The value of $x^2-6x+13$ can never be less than: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $

1969 IMO Longlists, 1

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

$(BEL 1)$ 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 $P_1.$

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$

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 $r + s$.

2001 AMC 12/AHSME, 13

Tags: parabola , geometry , geometric transformation , reflection , function , conic

The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$? $ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$

2024 All-Russian Olympiad Regional Round, 9.2

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

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.

1957 AMC 12/AHSME, 43

Tags: parabola , analytic geometry , conic

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $ x$-axis, the line $ x \equal{} 4$, and the parabola $ y \equal{} x^2$ is: $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \text{not finite}$

2009 Today's Calculation Of Integral, 521

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

Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.

2021 CCA Math Bonanza, L3.4

Tags: parabola , conic

Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$. [i]2021 CCA Math Bonanza Lightning Round #3.4[/i]

2009 Today's Calculation Of Integral, 397

Tags: calculus , integration , parabola , quadratic , function , analytic geometry , conic

In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis

2022 Saint Petersburg Mathematical Olympiad, 4

Tags: parabola , algebra

We will say that a point of the plane $(u, v)$ lies between the parabolas $y = f(x)$ and $y = g(x)$ if $f(u) \leq v \leq g(u)$. Find the smallest real $p$ for which the following statement is true: for any segment, the ends and the midpoint of which lie between the parabolas $y = x^2$ and $y=x^2+1$, then they lie entirely between the parabolas $y=x^2$ and $y=x^2+p$.

2014 AMC 12/AHSME, 17

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

Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$ $ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $

2024 ISI Entrance UGB, P7

Tags: maximal , parabola , applications of calculus , conic

Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.

2008 Harvard-MIT Mathematics Tournament, 26

Tags: parabola , ratio , geometry , geometric transformation , dilation , rotation , conic

Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.

2012 Today's Calculation Of Integral, 777

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

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

2006 AMC 10, 8

Tags: quadratic , parabola , conic

A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$? $ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$

2021 CCA Math Bonanza, L4.1

Tags: parabola , conic

Suppose that $x^2+px+q$ has two distinct roots $x=a$ and $x=b$. Furthermore, suppose that the positive difference between the roots of $x^2+ax+b$, the positive difference between the roots of $x^2+bx+a$, and twice the positive difference between the roots of $x^2+px+q$ are all equal. Given that $q$ can be expressed in the form $\frac{m}{m}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$. [i]2021 CCA Math Bonanza Lightning Round #4.1[/i]

2007 Today's Calculation Of Integral, 211

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

When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

2011 Today's Calculation Of Integral, 689

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

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

2006 China Second Round Olympiad, 13

Tags: parabola , conic

Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$.

2024 All-Russian Olympiad Regional Round, 10.2

Tags: algebra , parallelogram , quadratic , parabola , geometry , conic

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 lateral sides of all such trapezoids share a common point.

2013 Today's Calculation Of Integral, 874

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 $S$.

2011 Today's Calculation Of Integral, 684

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

On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$. [i]2011 Kyoto University entrance exam/Science, Problem 3[/i]