Problem Bank

Visitors to a landmark are surveyed one at a time about whether they plan to visit a second attraction. A visitor who declines scores $1$ point and a visitor who accepts scores $2$…

Real numbers $x, y, z$ satisfy

Fin…

Let $f(x) = x\ln x - 3x$.

1. Find the extreme values of $f(x)$. 2. If the inequality

holds for all $x$ in the domain, find the minimum…

Let $f(x) = \mathrm{e}^x - x - 1$.

1. Prove $f(x) \ge 0$ for all real $x$. 2. Prove that for every positive integer $n$,

Let $a, b > 0$ with

Find the range of possible values of $ab$.

Let $x, y, z \in (0, +\infty)$ with $x^2 + y^2 + z^2 = 1$. Find the range of

Points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ lie on the circle

Find the maximum value of

Find the sum of the integer parts of all positive real numbers $x$ satisfying

Let $f(x) = x^3 + bx + c$, and suppose the tangent to $y = f(x)$ at $x = \dfrac12$ is perpendicular to the $y$-axis.

1. Find $b$. 2. If $f$ has a zero of absolute value at most $1…

Inside equilateral $\triangle ABC$ there is a point $P$ with $PC > PB$. Given that $\angle BPC = 150^\circ$,…

Let $a$ and $b$ be real numbers, and let $f(x)=|a^2x^2-1|+ax$. Suppose that

Find the minimum value of

A function $f(x)$ defined on $(-1, 1)$ satisfies:

• for all $x, y \in (-1, 1)$, $\ f(x) + f(y) = f\left(\dfrac{x + y}{1 + xy}\right)$;
• $f(x) > 0$ whenever $x > 0$.

Determine, w…