Problem Bank

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$. A line $AB$ passes through the fixed point $Q(-1,0)$; let $P(1,0)$. The lines $PA$ and $PB$ meet the ellipse again at points $C$…

Let $\alpha, \beta \in \mathbb{R}$ satisfy

Find the minimum value of $\sin\alpha + \cos\beta - 2$.

Convex quadrilateral $ABCD$ has side lengths $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Its diagonals intersect at point $P$, and the sum of the areas of triangles $APB$ and…

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 …

The line $y = kx$ meets the curve $y = \ln x$ at two distinct points $M(x_1, y_1)$ and $N(x_2, y_2)$, and the tangent lines to $y = \ln x$ at $M$ and $N$ meet at $P(x_0, y_0)$. Det…

A cube $ABCD\text{-}A_1B_1C_1D_1$ has edge length $1$. A point $P$ moves over the surface of the cube with $PA = r$ (where $0 < r < \sqrt{3}$), and $f(r)$ denotes the total length …

Let

1. If the minimum value of $f(x)$ is $1$, find the range of possible values of $a$. 2. If the minimum value …

A sequence $\{a_n\}$ has $a_1 = 0$, and $a_n - a_{n-1} \in \{1, 2\}$ for every integer $n \geqslant 2$. Suppose a positive integer $k \in [1, 9]$ is such that

The Riemann zeta function $\zeta(s)$ is closely tied to the distribution of primes. Writing $\operatorname{Re}(s)$ for the real part of a complex number $s$, define

Let $f(x)=-2(x+a)\ln x+x^2-2ax-2a^2+a$, where $a>0$.

1. Let $g(x)$ be the derivative of $f(x)$. Discuss the monotonicity of $g(x)$. 2. Prove that there exists $a\in(0,1)$ such tha…

Let $F_1, F_2$ be the left and right foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{4} = 1$, let $A$ be a point on the right branch of $C$, and let $P\left(\sqrt{5}…

In the coordinate plane, the graph of the parabola $y = ax^2 - 3x + 3$ (with $a \neq 0$) is the reflection of the graph of the parabola $y^2 = 2px$ (with $p > 0$) across the line $…