Problem Bank

In a seminar of $n$ students, three problems $A$, $B$, $C$ are discussed. Every pair of students discusses exactly one of the three problems. Find the minimum value of $n$ that gua…

Consider the ellipse $C \colon \dfrac{x^2}{9} + \dfrac{y^2}{8} = 1$ and the circle $M \colon x^2 + 2x + y^2 + m = 0$. A line through the left vertex $A$ of the ellipse is tangent t…

A function on $[1, +\infty)$ is defined by

Determine, with proof…

In a regular triangular prism $ABC\text{-}A_1B_1C_1$ with all edges of length $1$, let $D$ and $E$ be the midpoints of $BB_1$ and $B_1C_1$. The cross-section through $A$, $D$, $E$ …

Positive reals $x, y$ satisfy

Find the minimum of $x + y$.

Consider the curve $C \colon ax^2 + by^2 + xy = 1$, where $a, b$ are constants. Determine, with proof, which of the following are true:

1. the curve $C$ is symmetric about the ori…

1. The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has right focus $F$; a line through $F$ meets the ellipse at $A, B$, and $C$ is the reflection of $A$ ac…

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

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

A moving point $M(x, y)$ has distance to the line $l \colon x = 4$ equal to twice its distance to the point $N(1, 0)$.

1. Find the equation of the locus $C$ of $M$. 2. A line $m$ …

A pentahedron $ABC\text{-}DEF$ has $AD \parallel BE \parallel CF$, with the three lines pairwise at distance $1$, and $AD = 1$, $BE = 2$, $CF = 3$. Find the volume of the pentahedr…

Let $f(x) = \ln x$.

1. If $g(x) = f(x) - ax + \dfrac{1}{2}x^2$ has two extreme points, find the range of possible values of $a$. 2. If the equation $f(x) = m(x+1)$ with $m \in \ma…

Let

and let $M = [a, b]$ with $a < b$, $N = \{y \mid y = f(x),\ x \in M\}$. If $M = N$, find $b - a$.