Problem Bank

The quadratic $f(x) = x^2 + bx + c$ passes through $(1, 13)$, and $y = f\left(x - \tfrac12\right)$ is even.

1. Find $f(x)$. 2. Are there points on $y = f(x)$ whose $x$-coordinate …

Consider the ellipse $G \colon \dfrac{x^2}{2} + y^2 = 1$. A line $l$ through the left focus $F$, not lying along the $x$-axis, meets $G$ at $A$ and $B$; let $M$ be the midpoint of …

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying

The distance from the point $M(3, 2)$ to the directrix of the parabola $C \colon y = ax^2$ (with $a > 0$) is $4$. Let $F$ be the focus of the parabola, let $N(1, 1)$, and let $P$ m…

As shown in the figure, an equilateral triangle $ABC$ has side length $2$. Its vertices $B$ and $C$ move along the…

The function $f(x) = ax + b$ satisfies $|f(x)| \leqslant 1$ for all $x \in [0, 1]$. Find the range of possible values of $(a+1)(b+1)$.

Suppose the equation

has two distinct real roots in the interval $[k-1,k+1]$. Find the range of the real number $k$.

For points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the coordinate plane, define the taxicab distance

Fix two distinct points $A$ and $B$.

1…

Find the largest positive integer $n$ with the property: however the numbers $1$ through $400$ are placed into a $20 \times 20$ grid, some row or column contains two numbers differ…

The function

has exactly one local maximum and one local minimum on $(0, 2\pi)$. Find the range of possible values of the real number $m$.

A cube has edge length $1$. Among all planes that cut the cube, find the maximum possible area of the resulting cross-secti…

Let $f(x) = \mathrm{e}^{\sin(x^2)}$. Find $f^{(8)}(0)$.