Problem Bank

Solve over the reals:

A set $A = \{a_1, \dots, a_n\}$ with $1 \le a_1 < a_2 < \cdots < a_n$ and $n \ge 2$ has property P: for all $1 \le i \le j \le n$, at least one of $a_i a_j$ and $\dfrac{a_j}{a_i}…

Let $f(x) = \ln x + (e - a)x - 2b$. If $f(x) \leqslant 0$ for all $x \in (0, +\infty)$, find the minimum value of $\dfrac{b}{a}$.

For every positive integer $n$ and real $\theta$, prove that

Let $f(x)$ satisfy

For $x>0$, determine, with justification, whether $f(x)$ has a local maximum and/or a lo…

In an acute triangle $ABC$ with orthocenter $H$, the feet of the three altitudes are $D,E,F$. Determine which notable…

A right triangle $PQR$ with legs $1$ and $2$ (hypotenuse $\sqrt5$) is inscribed in an isosceles right triangle $ABC$ whose right angle is at $C$, so that $AC=BC$ and $AB$ is the hy…

Let $f(x) = \dfrac{e^x - 1}{x}$.

1. Find the intervals of monotonicity of $f(x)$. 2. Suppose

and let $\lambd…

The sides $a, b, c$ of a right triangle satisfy

Find the maximum possible area of the triangle.

In parallelepiped $ABCD\text{-}A_1B_1C_1D_1$ the base is a rhombus, and the three edges meeting at vertex $C$ ma…

Sets $A, B, C$ satisfy $A \cup B \cup C = \{1, 2, \ldots, 2023\}$ and $A \cap B \cap C = \varnothing$. Let $n$ be the number of such ordered triples $(A, B, C)$. Find the last two …

Let $P$ be the bottom vertex of the ellipse $E \colon \dfrac{x^2}{4} + y^2 = 1$. Two mutually perpendicular lines $l_1, l_2$ pass through $P$: the line $l_1$ meets the circle $x^2 …