Problem Bank

A line through the fixed point $P(1, 1)$ meets the parabola $x^2 = 4y$ at $A$ and $B$. The tangent lines $\ell_1, \ell_2$ at $A$ and $B$ meet at $C$.

1. Find the equation of the l…

Let $f(x) = e^{x-a} + ax^2 - 3ax + 1$, where $a \in \mathbb{R}$.

1. When $a = 1$, find the equation of the tangent line to $y = f(x)$ at $x = 1$. 2. When $a > 1$, determine the nu…

In the first quadrant of the coordinate plane $xOy$, a point $P$ satisfies $OP=1$ and the line $OP$ has inclination $30^\circ$. An arbitrary line through $P$ meets the positive $x$…

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

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

As shown in the figure, the octahedron $PABCDQ$ is inscribed in a sphere with center $O$. The vertices $P$ and $Q$ lie on oppo…

Find the center of symmetry of the graph of

Let $f(x) = a^x + b^x - c^x$, where $c > a > 0$ and $c > b > 0$.

1. Let $M$ be the set of triples $(a, b, c)$ such that $a, b, c$ cannot be the side lengths of a triangle and $a =…

Prove that when $a > \mathrm{e}^{1/\mathrm{e}}$, the graphs of $f(x) = a^x$ and $g(x) = \log_a x$ have a common tangent line.

A square $ABCD$ has side length $30$. A point $P$ inside the square satisfies $AP=12$ and $BP=26$. The centroids of $\triangle ABP$, $\triangle BCP$, $\triangle CDP$, $\triangle DA…

Let $p$ and $q$ be positive integers, each at most $100$. How many polynomials of the form

have a rational root?

Let $P\text{-}ABCD$ be a regular quadrilateral pyramid with height $3$ and base edge $2$, and let $M$ be the midpoint of edge $PC$. A plane through $A$ and $M$ meets segments $PB$ …

Positive $x, y$ satisfy $xy(x+y) = 4$. Find the maximum of $xy$ and the minimum of $2x + y$.