Problem Bank

In a sequence $\{a_n\}$, $a_1 = 1$ and

where $c \neq 0$ is real.

1. Find a formula for $a_n$. 2. If $a_{2k} > a_{2…

Real numbers $a,b$ satisfy

Determine the sign of $a$ and the sign of $b$ (each $>0$ or $<0$).

Let $f(x) = \dfrac{\ln x + m}{x^2}$.

1. When $m = 1$, find the maximum value of $f(x)$. 2. Determine the number of real solutions of the equation $f(x) = m - \ln x$, discussing al…

Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\dfrac{x^2}3+y^2=1$. Points $A$ and $B$ lie on the ellipse with $\overrightarrow{F_1A}=5\overrightarrow{F_2B}$. Find …

In a seminar of $n$ students, three problems $A$, $B$, $C$ are discussed. Every pair of students discusses exactly one of the three problems. Find the minimum value of $n$ that gua…

Let $f(x) = e^{ax} - 1 + \ln(x+1)$.

1. If $f(x)$ is increasing on $(-1, +\infty)$, find the range of possible values of $a$. 2. If $0 < a \leqslant 1$ and $x > 0$, prove that $f(x…

In a regular triangular frustum $ABC\text{-}A_1B_1C_1$, $AB = 2A_1B_1 = 2\sqrt{3}$ and $AA_1 = 2$. Let $D$ be the midpoint of edge $AB$, and let $l$ be a moving line in the plane $…

Let $S_n$ be the partial sums of a sequence $\{a_n\}$ with $a_1 = 1$, $a_2 = 6$, $a_3 = 11$, and

where $A, B$…

In $\triangle ABC$, $\angle A=60^\circ$, and $\angle BAP=\angle CAP$ with $P$ inside the triangle. The extension of $BP$ meets $AC$ at $Q$, and

Find the least real $a$ such that

holds for all $x, y$ with $|x| + |y| \le 1$.

Consider the ellipse $\dfrac{x^2}{4} + y^2 = 1$ and the fixed point $P(4, 0)$. A line of slope $1$ meets the ellipse at $A$ and $B$; lines $PA$ and $PB$ meet the ellipse again at p…

A cube $ABCD\text{-}A_1B_1C_1D_1$ has edge length $2$. The vertices $A_1, B_1, C_1, D_1$ lie on the lateral surface of a cone, and the vertices $A, B, C, D$ lie in the plane of the…