Problem Bank

Let $f(x) = x^3 - ax^2 + \left(a^2 - 2\right)x + 1$. If there exists $m > 0$ with $f(m) \leqslant 0$, find the maximum possible value of the real number $a$.

A player enters a best-of-three match (first to win two games wins the match). Each game is won with probability $p$ ($0 < p < 1$) and lost with probability $1 - p$, independently …

Two players compete in a series of games; in each game, player $1$ wins with probability $p_1$ (where $0 < p_1 < 1$) and player $2$ wins with probability $p_2 = 1 - p_1$. The first…

As shown in the figure, $P$ is a point outside the cube $ABCD$-$A_1B_1C_1D_1$. A line $l$ is drawn through $P$; let $\…

The function

has both a local maximum and a local minimum in the interval $(0, 1)$. Find the range of

An equilateral triangle $ABC$ has side length $1$ and center $O$. A line through $O$ meets the sides $AB$ and $AC$ at $M$ and $N$ respectively. Find the range of possible values of…

Let $a \in (0, 1)$, and define

1. Prove that $f(x)$ has exactly $2$ zeros. 2. For $x \in (0,…

A moving line parallel to the $x$-axis meets the hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) at points $A$ and $B$, with $A$ to the left of $B$.…

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$. A point $A$ lies on $C$ and a point $B$ lies on the $y$-axis, with

Let $f(x)=ax^2+bx+c$ ($a,b,c\in\mathbb R$, $a>0$). Let $p$ be the statement "both $f(x)$ and $f(f(x))$ have exactly two zeros," and $q$ the statement "$f!\left(f!\left(-\dfrac b{…

The cubic $f(x) = ax^3 + bx^2 + cx + d$ (with $a < b$) is increasing on all of $\mathbb{R}$. Find the minimum value of

Let $P$ be an intersection point of the circles

where $ac = 8$ and $\dfrac{a}{b…