Problem Bank

Real numbers $x, y$ satisfy

Find the minimum value of

Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\dfrac{x^2}3+\dfrac{y^2}2=1$. A line through $F_1$ meets the ellipse at two points $A$ and $B$, with $|AF_2|=|AB|$ an…

Find the range of the function

In the expansion of $(a+b)^n$, some three consecutive binomial coefficients form an arithmetic progression. Find the largest three-digit positive integer $n$ for which this happens…

Let $O$ be the circumcenter of $\triangle ABC$, with

Find the angle $C$.

Let $P$ be an intersection point of the circles

where $ac = 8$ and $\dfrac{a}{b…

Prove that for every nonzero real number $x$,

Let $f(x) = (1 - ax)\ln(1 + x) - x$.

1. When $a = -2$, find the extreme values of $f(x)$. 2. If $f(x) \geqslant 0$ for all $x \geqslant 0$, find the range of possible values of $a…

A line $\ell$ meets the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{2} = 1$ at $A$ and $B$, the negative $x$-axis at $C$, and the positive $y$-axis at $D$, with $A, C, D, B$ in order alo…

Let $C$ be a hyperbola with center $O$. Suppose there is exactly one pair of lines $A_1B_1$ and $A_2B_2$ through $O$, making an angle of $60^\circ$ with each other, such that $|A_1…

Suppose

with $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

Consider the hyperbola $\dfrac{x^2}{2} - y^2 = 1$ with fixed points $A(-2, 1)$ and $B(2, 1)$ on it. A line through $M(0, -3)$ meets the hyperbola at $C$ and $D$, and lines $AC$ and…