Problem Bank

Let

where $a$ is real.

1. When $a = 1$, find the intervals of monotonicity of $f(x)$. 2. If $f(x) \leqslant 0$ for all $-1 < x < 1$, find t…

The function

has a local maximum in $(0, 1)$ and a local minimum in $(1, 2)$. If $(a + 3)b \leqslant t$ holds for all such $…

In $\triangle ABC$ (usual notation),

1. Find $A$. 2. If $a = \sqrt3$ and $I$ is the incenter, find the maximum of $IB + IC$.

An infinite sequence $P \colon a_1, a_2, \ldots$ satisfies $a_i \in \mathbb{N}^*$ and $a_i \leqslant a_{i+1}$. For $k \in \mathbb{N}^*$, let $T_k(P)$ be the smallest element of ${…

In $\triangle ABC$, the perpendicular bisector of the side $BC$ meets $BC$ at $Q$ and meets $AC$ at $P$. Given $\left|\overrightarrow{AB}\right|=1$ and $\left|\overrightarrow{AC}\r…

Let $M = \{1, 2, \dots, 10\}$ and

How many elements does $A$ have?

Let

and suppose $f(x_1) = f(x_2)$ with $x_1 \neq x_2$. Prove that $x_1 + x_2 > 2$.

Let $f(x)$ have domain $D$. If there exists $[a, b] \subseteq D$ such that the minimum of $f(x)$ on $[a, b]$ is $\dfrac{a}{n}$ and the maximum is $\dfrac{b}{n}$, where $n$ is a pos…

Let $a, b, c$ be positive integers and $f(x) = ax + \dfrac{c}{x}$. The equation $f(x) = b$ has two real solutions in $(0, 1)$, and exactly $2$ values of $b$ have this property. Fin…

Let

If the minimum value of $f(x)$ is $0$, find $t$.

The function

has exactly two centers of symmetry with $x$-coordinates in the interval $\left[\dfrac{…

Consider the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ and its left vertex $A(-a, 0)$. A line $l$ through $A$ meets the ellipse again at $Q$ and meets the $y$-axis at $R$; …