Problem Bank

The figure is made of six squares: three inner squares are erected outward on the sides of a central triangle, and each outer sq…

A line $l$ meets the parabola $y^2 = 4x$ at two points $A$ and $B$. If the quadrilateral $OAMB$ (where $O$ is the origin) is a rectangle, find the maximum value of the slope of the…

A fair coin is tossed repeatedly; heads scores $1$ point, tails scores $2$ points. Find the probability that the running total ever equals exactly $n$.

Let $a$ be a constant, and suppose $f(x) = x(\ln x - ax)$ has two critical points $x_1 < x_2$. Determine, with proof, the sign of $f(x_1)$, and how $f(x_2)$ compares with $-\dfrac{…

The graph of a cubic function $f(x)$ passes through the points $A(2, 4)$, $B(3, 9)$, $C(4, 16)$. Lines $AB$, $AC$, $BC$ meet the graph of $f(x)$ again at points $D$, $E$, $F$ respe…

Let $A$ be the right vertex of the hyperbola $C:x^2-\dfrac{y^2}{b^2}=1$ ($b>0$). A line $l$ through the point $T(0,t)$ meets the left and right branches of $C$ at points $M$ and $N…

A geometric sequence $\{a_n\}$ has first term $a_1 > 0$ and common ratio $q > 1$. Define

If $l_n$ is …

Nonnegative reals $x, y$ satisfy $x + y = 4$. Find the minimum of

Let $n > 2$ and let $a_1, \dots, a_n \in (0, 1]$. With $A_k = \dfrac{a_1 + \cdots + a_k}{k}$, prove that

A sequence $\{a_n\}$ satisfies $a_1 = 1$, $a_2 = 2$, and for $k \in \mathbb{N}^+$,

Find the last t…

For a given positive integer $n$ ($n \geqslant 2$), let $M = \{k \in \mathbb{Z} \mid -n \leqslant k \leqslant n\}$. Nonempty sets $A, B$ satisfy $A \cup B = M$ and $A \cap B = \var…

Prove that there are infinitely many odd positive integers $m$ for which $8^m + 9m^2$ is composite.