Problem Bank

The function

has exactly one extreme point. Find the range of possible values of the real number $a$.

Let $a, b, c > 0$ with

Find the maximum value of

A sequence $\{a_n\}$ satisfies $a_0 = 0$ and

Prove that every term of the sequence is a n…

Let $f(n)$ be the integer nearest to $\sqrt{n}$. If

find the positive integer $m$.

As shown in the figure, let $F$ be the focus of the parabola $y^2 = 4x$, and let $AB$ and $CD$ be two chords of the par…

A sequence $a_1, a_2, \ldots, a_{21}$ satisfies $|a_{n+1} - a_n| = 1$ for $n = 1, \ldots, 20$, and $a_1, a_7, a_{21}$ form a geometric sequence. If $a_1 = 1$ and $a_{21} = 9$, find…

Let $z$ be a non-real complex number with $|z| \ne 1$, and set

Given that $a = w + \dfrac{1}{w}$ is a real number, find the range of $a$.

A sequence has $a_1 = \dfrac{1}{2}$ and

and $b_n = na_n$.

1. Prove that $\{a_n\}$ is decreasing. 2. Prove that $b_n \leqslant \dfrac{3}…

A regular $2020$-gon $A_1A_2\cdots A_{2020}$ is inscribed in the unit circle $O$. Two distinct vertices $A_i, A_j$ are chosen at random.

1. Find the probability that $\overrightar…

As shown in the figure, in the cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $1$, a point $P$ moves on the segment $A_…

Six students sit in a row of connected desks and can leave only from the two ends of the row. They finish their exams in a uniformly random order, and a student who wants to leave …

Let $a_n$ be the integer nearest to $\sqrt{n}$. Find