Problem Bank

The ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$ has foci $F_1$ (left) and $F_2$ (right). A line $l$ with nonzero slope meets the ellipse at $A$ and $B$ ($A$ to the left of $B$), …

Let $P$ move along the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) with foci $F_1, F_2$, and let $I$ be the incenter of $\triangle PF_1F_2$. Find …

The lateral surface of a cone unrolls into a circular sector of fixed area with central angle $\alpha$. When the volume of the sphere inscribed in the cone is largest, find $\alpha…

Solve over the reals:

A regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. A point $P$ inside the circle is such that the region bounded by $PA_1$, $PA_2$, and the minor ar…

A line through the fixed point $(1,0)$ meets the parabola $y^2=2x$ at two points $A$ and $B$. Let $C$ be the point for which the origin $O$ is the centroid of $\triangle ABC$ (that…

Real numbers $a, b, c, d$ satisfy

Find $(a+c)(b+c)$.

An octagon $ABCDEFGH$ is obtained from a $23 \times 27$ rectangle by cutting off a right triangle with side lengths $6$, $…

In the coordinate plane, a moving point $E$ is equidistant from the fixed point $(1, 0)$ and the line $x = -1$.

1. Find the equation of the locus $C$ of $E$. 2. A moving line $l \…

Let $E = \{1, 2, 3, \dots, 200\}$ and let $G = \{a_1, a_2, \dots, a_{100}\}$ be a $100$-element subset of $E$ with the following two properties:

1. for all $1 \leqslant i \leqslan…

A set $A = \{a_1, a_2, \ldots, a_{50}\}$ consists of $50$ distinct positive integers. Let $S(A)$ be the sum of all elements of $A$ and $E(A)$ the number of even elements. If $S(A) …

Let $f(x) = x \ln x$.

1. Let $g(x) = \dfrac{f'(x)}{x}$. Find the extreme points of $g(x)$. 2. If for all $x_2 > x_1 > 0$ we have