Problem Bank

Define $f(x) = \dfrac{4^x}{4^x + 2}$ on $\mathbb{R}$, and for $n = 2, 3, \ldots$ set

Eleven students volunteer as museum guides. From 9 am to 5 pm there is one talk each hour ($8$ talks in total), and each talk needs exactly one student speaker. To avoid fatigue, n…

Let $a = \dfrac{3}{2}$, and let $b, c$ satisfy $3^b = 5$ and $5^c = 8$. Arrange $a, b, c$ in increasing order, with proof.

Let $f_{k,t}(x)=\dfrac{kx+t}x$ (for $k,t,x\in\mathbb R$, $x\neq 0$).

1. If $f_{1,2}(1)$, $f_{2,2}(x)$, $f_{1,3}(3)$ form an arithmetic progression, find $x$. 2. Suppose $\left{f_…

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

The graph of $f(x) = x^3 + px^2 + qx$ is tangent to the $x$-axis at a point $(x_0, 0)$ with $x_0 \neq 0$, and the local minimum value of $f$ is $-4$. Find $p + q$.

Real numbers $x, y, z$ satisfy

Fin…

For a positive integer $n$ and reals $a_0, \dots, a_n$, let

1. If $a_0 = 0$ and $a_1 =…

The equation $x^3 - 3x + 4 = 0$ has three roots $a, b, c$ (counted with multiplicity, possibly complex). Find the value of

The arithmetic sequence $\{a_n\}$ has partial sums $S_n$ with $S_{10} = 0$ and $S_{15} = 25$. Find the minimum value of $nS_n$.

Let $p$ and $q$ be positive integers, each at most $100$. How many polynomials of the form

have a rational root?

Let $m, n \geqslant 0$, and suppose

is decreasing on the interval $\left[\dfrac{1}{2}, 2\right]$. Find the maximum value of $mn$.