Problem Bank

Let $\{a_n\}$ be an infinite sequence of nonnegative integers. Let $A_n$ denote the maximum of the first $n$ terms, and $B_n$ the minimum of the terms after the $n$th: $B_n = \min\…

For $n = 1, 2, \dots$, define

1. Prove that for each $n \in \mathbb{N}^*$ there exists a uniq…

Let $f_t(x)=(x-t)^2-t$ ($t\in\mathbb R$), and for $a<b$ define

If the function $y=f(x)+x+a…

The set $A = \{1011, 1012, \dots, 2022\}$ is partitioned arbitrarily into two disjoint nonempty subsets. Prove that at least one of the subsets contains two numbers whose sum is a …

Positive reals $x, y, z$ satisfy

Find the maximum value of $xy + 2xz$.

Suppose

Determine which of the values

Let $f(x) = x - x^3\mathrm{e}^{ax+b}$, and suppose the tangent line to $y = f(x)$ at the point $(1, f(1))$ is $y = -x + 1$.

1. Find $a$ and $b$. 2. Let $g(x) = f'(x)$. Find the in…

A function on $[1, +\infty)$ is defined by

Determine, with proof…

Arrange $n$ ($n \geqslant 2$) distinct numbers in some order to form a sequence $\{a_n\}$. For $1 \leqslant i < j \leqslant n$, if $a_i > a_j$, the pair $(a_i, a_j)$ is called an *…

The equation

has exactly one real solution in $(0, +\infty)$. Find the range of $a$.

Let $A(2,1)$ be a fixed point on the hyperbola $\dfrac{x^2}2-y^2=1$, and let $P,Q$ be two points on the hyperbola such that the slopes of $AP$ and $AQ$ are negatives of each other.…

The upper bound $M$ of the state-space complexity of the game of Go is about $3^{361}$, and the number of atoms of ordinary matter in the observable universe is about $N\approx 10^…