Olympiad problems

1952 AMC 12/AHSME, 13

Tags: quadratic , function , parabola , calculus , conic

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$

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.

2014 HMNT, 6

Tags: combinatorial geometry , geometry , parabola , conic

Let $P_1$, $P_2$, $P_3$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $P_i$. In other words, find the maximum number of points that can lie on two or more of the parabolas $P_1$, $P_2$, $P_3$ .

2011 Today's Calculation Of Integral, 731

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

Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively. When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.

2005 AMC 12/AHSME, 23

Tags: parabola , logarithm , conic

Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

1999 Federal Competition For Advanced Students, Part 2, 3

Tags: hyperbola , parabola , function , probability , floor function , conic

Find all pairs $(x, y)$ of real numbers such that \[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\] where $f(x)=[x]$ is the floor function.

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$

2024 Assara - South Russian Girl's MO, 4

Tags: geometry , algebra , parabola , conic

A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane. [i]P.A.Kozhevnikov[/i]

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 $h$.

2016 AMC 12/AHSME, 6

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 $BC$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

2008 Paraguay Mathematical Olympiad, 2

Tags: parabola , absolute value , conic

Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$

2011 AMC 12/AHSME, 14

Tags: trigonometry , parabola , vector , pythagorean theorem , trig identities , conic , geometry

A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$? $ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2} $

2009 Today's Calculation Of Integral, 492

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

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.

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) $N = 2012$ ?

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 $T'$.

2007 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , parabola , function , conic

Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.

2011 AMC 12/AHSME, 14

Tags: probability , parabola , inequalities , conic

Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? $ \textbf{(A)}\ \frac{11}{81} \qquad \textbf{(B)}\ \frac{13}{81} \qquad \textbf{(C)}\ \frac{5}{27} \qquad \textbf{(D)}\ \frac{17}{81} \qquad \textbf{(E)}\ \frac{19}{81} $

2007 Moldova Team Selection Test, 1

Tags: parabola , symmetry , ellipse , hyperbola , geometry , conic

Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.

2011 Romanian Master of Mathematics, 3

Tags: geometry , circumcircle , geometric transformation , trigonometry , parabola , conic

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

1981 All Soviet Union Mathematical Olympiad, 308

Tags: geometry , rectangle , minimum , parabola , area

Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$

2014 AMC 12/AHSME, 25

Tags: parabola , calculus , derivative , vector , analytic geometry , parameterization , conic

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

1998 National High School Mathematics League, 15

Tags: parabola , analytic geometry , conic , geometry

Parabola $y^2=2px$, two fixed points $A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)$. $M$ is a point on the parabola, $AM$ intersects the parabola at $M_1$, $BM$ intersects the parabola at $M_2$. Prove: When $M$ changes, line $M_1M_2$ passes a fixed point, and find the fixed point.

2012 AMC 12/AHSME, 13

Tags: parabola , probability , symmetry , conic

Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common? $\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $

1962 AMC 12/AHSME, 15

Tags: parabola , ellipse , analytic geometry , ratio , conic

Given triangle $ ABC$ with base $ AB$ fixed in length and position. As the vertex $ C$ moves on a straight line, the intersection point of the three medians moves on: $ \textbf{(A)}\ \text{a circle} \qquad \textbf{(B)}\ \text{a parabola} \qquad \textbf{(C)}\ \text{an ellipse} \qquad \textbf{(D)}\ \text{a straight line} \qquad \textbf{(E)}\ \text{a curve here not listed}$

2018 Belarusian National Olympiad, 9.5

Tags: geometry , parabola , conic

The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD. Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$.