Problem Bank

A sequence $\{a_n\}$ has exactly $m^2$ terms (with $m > 2$). The first block $a_1, a_2, \dots, a_m$ is an arithmetic sequence with nonzero common difference, and for each $i \in {…

On a segment of length $n$ ($n \in \mathbb{N}^*$), points are marked at every unit distance, giving $n + 1$ points including the endpoints.

1. How many segments have both endpoint…

Consider the following list of sequences:

• Sequence $1$: $\quad 1, 1, 1, 1, 1, 1, 1, 1, \dots$
• Sequence $2$: $\quad 1, 2, 3, 4, 5, 6, 7, 8, \dots$
• Sequence $3$: $\quad 1, 2, …

Consider the parabola $E \colon y = x^2$. A line through $T(1, 2)$ meets $E$ at two points $A$ and $B$. Let $l_1$ and $l_2$ be the tangent lines to $E$ at $A$ and $B$; $l_1$ meets …

As shown in the figure, the ellipse $C_1:\dfrac{x^2}{4}+y^2=1$ and the parabola $C_2:x^2=2py$ (where $p>0$) intersect a…

The parabola $C \colon y^2 = 2px$ ($p > 0$) has focus $F$, and the point $P(4, t)$ on $C$ (with $t > 0$) is at distance $5$ from $F$.

1. Find $p$ and $t$. 2. Let $A, B$ be points …

In an acute triangle $ABC$,

Find the maximum value of $\cos C$.

Prove that the sequence

converges.

Let $f(x)=nx-x^n$ for $x\in\mathbb R$, where $n\in\mathbb N^{*}$ and $n\geqslant 2$.

1. Discuss the monotonicity of $f(x)$. 2. Let $P$ be the intersection point of the curve $y=f(…

Two players take turns removing pebbles from a pile of $15$; each turn removes $1$, $2$, or $3$ pebbles, and play ends when the pile is empty. The player who has taken an odd total…

In the coordinate plane, let $A(a, a)$ be a fixed point and let $P$ be a moving point on the graph of $y = \dfrac{1}{x}$ for $x > 0$. If the shortest distance between $P$ and $A$ i…

Let $f(x) = 8x^3 + ax^2 + bx$. Do there exist real numbers $a, b$ such that $|f(x)| \leqslant 2$ for all $x \in [-1, 1]$? If so, find all such $a, b$; if not, explain why.