Problem Bank

As shown in the figure, in the pentahedron $ABCDEF$ the quadrilateral $ABCD$ is a square, the plane $ADE$ is perpen…

Let $A \colon a_1, a_2, \ldots, a_{2m}$ be a permutation of $1, 2, \ldots, 2m$, where $m \in \mathbb{N}^*$ and $m \geqslant 3$. Say $A$ has property $P$ if there is at least one …

Consider the ellipse $\dfrac{x^2}4+y^2=1$ and the hyperbola $\dfrac{x^2}4-y^2=1$. From the left vertex $P(-2,0)$, two lines are drawn meeting the ellipse at $A$, $C$ and the hyperb…

Real numbers $a,b$ satisfy

Determine the sign of $a$ and the sign of $b$ (each $>0$ or $<0$).

Say a sequence $\{a_n\}$ has property $P(t)$ if

1. If $a_n = 2^n$ has property $P(t)$, find …

Let $f(x)=\sin x$. Suppose there exist $x_1,x_2,\dots,x_m$ with $0\leqslant x_1<x_2<\cdots<x_m\leqslant 6\pi$ such that

In a right triangle $ABC$ with $\angle ACB=90^\circ$, $BC=a$, and $AC=b$, points $D,E,F$ lie on the sides $BC$, $CA$, $AB$ respectively so that $\triangle DEF$ is equilateral. Find…

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

In $\triangle ABC$, the angles satisfy

1. Does the value of $y$ change if any two of the angles $A, B, C$ are exchanged? Prove your con…

Let $f(x) = (a + \sin x)(a + \cos x)$, and let $g(a)$ denote the maximum value of $f(x)$ as $x$ ranges over the reals. Find $g(a)$, and find the minimum value of $g(a)$.

Let $S(x)$ be the digit sum of the natural number $x$. Solve

An equilateral triangle $ABC$ has incircle of radius $2$ centered at $I$. A point $P$ satisfies $PI=1$. Find the maximum value of the ratio of the areas of $\triangle APB$ and $\tr…