Problem Bank

Let $a,b,c\in\mathbb R$, and consider $f(x)=ax^2+bx+c$ and $g(x)=ax+b$, both on $-1\leqslant x\leqslant 1$. Prove that if $|f(x)|\leqslant 1$ for all $x\in[-1,1]$, then $|g(x)|\leq…

Given $|\vec a| = |\vec a + 2\vec b| = 2$, find the maximum of

Find the difference between the longest and shortest diagonals of a regular nonagon with side length $1$.

The sides $a, b, c$ of $\triangle ABC$ (opposite $A, B, C$) satisfy

Find $B$.

Let

Determine, with proof, which of the following claims are true.

1. When $a=1$, $f(x)$ has exactly one zero.…

An engineer uses a detector, with effective range $\dfrac{\sqrt 3}2$ km, to sweep for mines in an equilateral triangular region of side length $2$ km. Starting from one vertex of t…

The hyperbola $E \colon \dfrac{x^2}{4} - y^2 = 1$ meets the line $l \colon y = kx - 3$ at two points $A$ and $B$, and $M$ is the midpoint of segment $AB$.

1. As $k$ varies, find t…

In the coordinate plane, the point $A$ lies in the first quadrant (Fig. 1). The foot of the perpendicular from $A$ to…

Let $f(x) = (2-x)\mathrm{e}^{k(x-1)} - x$ with $k$ real.

1. If $f$ is decreasing on all of $\mathbb{R}$, find the maximum value of $k$. 2. For $x \in (1, 2)$, prove that

Let $[x]$ and $\{x\}$ denote the integer and fractional parts of $x$.

1. Find the number of zeros of $f(x)=\sin^2[x]+\sin^2\{x\}-1$ on $[0,100]$. 2. Find the number of zeros of $g…

Let $x,y,z$ be positive reals. Prove that

A real function $f$ satisfies

Do there exist integers $n$ with $f(n)$ also an integer? Find all such $n$ or prove none exist.