Problem Bank

Let $F_1, F_2$ be the left and right foci of the hyperbola $C \colon \dfrac{x^2}{2} - \dfrac{y^2}{2} = 1$. A line $l$ through $T(-3, 0)$ meets the left and right branches of $C$ at…

Let $a,b,c\in[0,1]$. Find the range of

Let $a, b, c, d \geqslant 0$ with $a + b + c + d = 4$. Find the maximum and minimum values of

As shown in the figure, the cube $ABCD\text{-}A_1B_1C_1D_1$ has edge length $1$. Points $E$ and $F$ move along the edges $…

Consider rational numbers $r = \dfrac{p}{q} \in (0, 1)$, where $p$ and $q$ are coprime positive integers and $pq$ divides $3600$. How many such rational numbers $r$ are there?

In $\triangle ABC$, $AB = AC$. Extend $AB$ to $D$ and extend $CA$ to $E$, and connect $DE$, so that $AD = BC = …

A sequence $\{a_n\}$ of positive terms has partial sums $S_n$, and for all $m, n \in \mathbb{N}^*$,

1. Find $\dfrac{a_2}{a_…

As shown in the figure, two congruent equilateral triangles $ABC$ and $A_1B_1C_1$ together form a six-pointed star with c…

Let $f(x) = ax^2 + 2x + 1$. If

find the range of possible values of the real number $a$.

The function $f(x) = -\dfrac{1}{2}x^2 + \dfrac{13}{2}$, restricted to an interval $[a, b]$, has minimum value $2a$ and maximum value $2b$. Find the interval $[a, b]$.

Let $O$ be the circumcenter of $\triangle ABC$, whose sides $a, b, c$ (opposite $A, B, C$) satisfy

Find the range of possible values of $\overrightarrow{B…

Seven points lie on a circle, and all segments joining pairs of them are drawn. Two of these segments are chosen at random. Find the probability that they have no common point (no …