Olympiad problems

2012 AIME Problems, 6

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.

2024 ELMO Shortlist, G8

Tags: geometry , ellipse , conic

Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$. [i]Andrew Carratu[/i]

2021 China Second Round Olympiad, Problem 12

Tags: ellipse , analytic geometry , conic

Let $C$ be the left vertex of the ellipse $\frac{x^2}8+\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$. (i) Compute the maximum of $|CA|+|CB|$. (ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and $N$, respectively. If the intersection of the perpendicular bisector of $MN$ and the circle with diameter $MN$ lies inside the given ellipse, compute the range of possible values of $k$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 12)[/i]

1997 AIME Problems, 7

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

A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

2007 Princeton University Math Competition, 8

Tags: geometry , calculus , integration , ellipse , ratio , conic

What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?

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.

1956 AMC 12/AHSME, 29

Tags: geometry , parallelogram , rectangle , rotation , hyperbola , symmetry , conic

The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is: $ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$ $ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$

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 $

1999 Turkey Team Selection Test, 3

Tags: parabola , combinatorics , conic

Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)

1940 Putnam, B3

Tags: analytic geometry , parabola , conic

Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$. ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$. iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$. iv) All three coincide only if $a=2p$ and $b=0$.

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]

1994 Cono Sur Olympiad, 2

Tags: quadratic , number theory , greatest common divisor , calculus , integration , analytic geometry , conic

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$