Problem Bank

Let $f(x) = (x + a)\ln(x + b)$. If $f(x) \geqslant 0$ for all $x$ in the domain, find the minimum value of $a^2 + b^2$.

Let $F_1, F_2$ be the foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a, b > 0$), and let $P$ be an intersection point of $C$ with the circle $x^2 + y^2 …

A function $f$ satisfies, for all $x, y \in \mathbb{R}$,

Write down one such function $f(x)$.

A sequence has $a_1 = 1$ and

Evaluate

Finitely many positive integers satisfy condition $T$: each is at most $7$ and their total sum is $S = 430$. They are divided into $M$ groups, each with sum at most $21$, by the fo…

An ellipse $C$ is centered at the origin with foci on the $x$-axis; its right focus is $F$, the line $l \colon x = 4$ is its right directrix, and the distance from $F$ to $l$ is $3…

The four vertices of a (not necessarily planar) quadrilateral $ABCD$ lie on a sphere $O$. Let $E, F$ be the midpoints of $AB$ and $CD$, with $EF \perp AB$ and $EF \perp CD$. If $AB…

In the coordinate plane, a parabola $C$ has vertex at the origin, is symmetric about the $x$-axis, and passes through $P(1, 2)$.

1. Find the equation of the parabola $C$. 2. Point…

A positive sequence $\{a_n\}$ satisfies $a_1 = 1$ and

Let $T_n = a_1a_2 + a_…

Let $a, b > 0$ with $a^2 - b + 4 \leqslant 0$, and set

Determine, with proof, whether $u$ attains a maximum and/or a minimum, and find any such ext…

The quadratic $f(x) = ax^2 + 2bx + c$ with $c > b > a$ passes through $(1, 0)$ and meets the line $y = -a$. Prove that

Let $b > 0$, and let the sequence $\{a_n\}$ satisfy $a_1 = b$ and

1. Find a formula for $a_n$. 2. Prove that…