Problem Bank

An operation $\&$ on real numbers satisfies, for all reals $x, y, z$,

Compute $2000 \& 2022$.

The ellipse $\Gamma: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has top vertex $C(0, 1)$, and the circle centered at $C$ with radius $\dfrac{4\sqrt3}{3}$ is tangent to …

Real numbers $a, b$ satisfy

Find the range of possible values of $a^2 + b^2$.

Find all pairs of positive integers $(x, y)$ such that

The function

has an extreme value on the interval $[t, +\infty)$, where $t \in \mathbb{N}^*$. Find the maximum possible value of $t$.

In triangle $ABC$, the sides opposite angles $A, B, C$ are $a, b, c$, and

If the bisector $AD$ of angle $A$ has length $2$, fin…

A sequence $\{x_n\}$ has $x_1 = 1$ and

Find $\displaystyle\sum_{n=1}^{2008} x_n$.

Prove that

Let $f(x) = ax^2 + 4x - 2$. Suppose the real number $a$ can be chosen so that

Find the minimum possible value of $m$, …

In the coordinate plane, let $A(-1, 0)$, and let $B, C$ move along the unit circle centered at the origin $O$ so that the line $BC$ has inclination $30^\circ$. Find the range of po…

Let $\{a_n\}$ and $\{b_n\}$ be two distinct infinite sequences, neither constant. Define $M = \{k \mid a_k = b_k,\ k \in \mathbb{N}^*\}$. Determine which of the following statement…

In the coordinate plane, let $A(-1, 0)$, $B(1, 0)$, and $P(x_0, y_0)$, where triangle $PAB$ satisfies

1. Can $(0, -1)$ be the …