Problem Bank

Call $f(x)$ convex on $(a, b)$ if it is differentiable there and $f'(x)$ is increasing.

1. Decide whether $y = x^3$ and $y = \ln\dfrac{1}{x}$ are convex on their domains. 2. Let…

The parabola $C:y^2=2px$ ($p>0$) has focus $F$. Let $A$ be any point on $C$ other than the origin; a line $l$ through $A$ meets $C$ again at $B$ and meets the positive $x$-axis at …

From each point of the circle $x^2+y^2=4$, the two tangent lines to the ellipse $C:\dfrac{x^2}2+y^2=1$ are drawn; the segment joining the two points of tangency is called a contact…

Let $f(x) = \dfrac{a}{2}\ln(x+1) - \sqrt{x+2}$, where $a \in \mathbb{R}$.

1. When $a = \dfrac{8}{3}$, find the intervals on which $f(x)$ is monotonic. 2. If for all $x \geqslant 0…

In the coordinate plane $xOy$, $O$ is the centre of a regular octagon $A_1A_2\cdots A_8$ with $A_1=(1…

Let $a,b\in\mathbb R$ with $a\neq 0$. The curves

have at least one common point with $x$-coordinate in the interval $[3,4]$. Fin…

Let $S = \{1, 2, 3, \dots, 10\}$. How many subsets $A$ of $S$ satisfy both

A right triangle $PQR$ with legs $1$ and $2$ (hypotenuse $\sqrt5$) is inscribed in an isosceles right triangle $ABC$ whose right angle is at $C$, so that $AC=BC$ and $AB$ is the hy…

Let $a, b, c \in \mathbb{R}$, and suppose

Find the maximum possible value of $|a\sin…

Positive reals $x, y, z$ satisfy $xy + yz + zx \ne 1$ and

Find

A dissection graph of a convex $n$-gon is the figure formed by the $n$-gon together with $n-3$ of its diagonals, no…

Let

If $m \neq n$ and $f(m) + f(n) = 4$, find the minimum value of $m + n$.