Find all triangles whose sides are consecutive integers and areas are also integers.
Find all positive integers $k$, $m$ such that $k < m$ and
$$1+ 2 +\cdots+ k = (k +1) + (k + 2) +\cdots+ m$$
Prove that there are infinitely many positive integers $n$ such that $(n^2+1)$ divides $n!$.
Let $x$ be a real number between 0 and 1. Find the maximum value of $x(1-x^4)$.
Prove: any convex pentagon must have three vertices $A$, $B$, and $C$ satisfying $\angle{ABC} \le 36^\circ$.
There are $99$ points on a plane. Among any three points, at least two of them are not more than $1$ unit length apart. Prove: it is possible to cover $50$ of these points using a unit circle.
If $a_0, a_1,\cdots, a_n \in \{0, 1, 2,\cdots, 9\}, n\ge 1, a_0\ge 1$, then the zeros of $f(x)=a_0 x^n + a_1x^{n-1} +\cdots +a_n$ have real parts less then 4.
Find all pairs $(a,b)$ of nonnegative reals such that $(a-bi)^n = a^n - b^n i$ for some positive integer $n>1$.
Find the number $x = [1, 2, 3, 1, 2, 3, \cdots]$. (continued fraction)
Find all nonnegative integers $x$ and $y$ such that $x^3+y^3 = (x+y)^2$.
Let unit vectors $a$, $b$, and $c$ satisfy $a+b+c=0$, prove the angles between these vectors are all $120^\circ$.
Let complex numbers $a$, $b$, and $c$ satisfy $a|bc| + b|ca| + c|ab| = 0$. Show that $$|(a-b)(b-c)(c-a)|\ge 3\sqrt{3}|abc|$$
Let $x, y \in \big(0, \frac{\pi}{2}\big)$. Show that if the equation $(\cos x + i \sin y)^n = \cos nx + i \sin ny$ holds for two consecutive positive integers, then it will hold for all positive integers.
Find all polynomials $f(x)$ such that $f(x^2) = f(x)f(x+1)$.
Let $z$ be a complex number and $k$ be a known real number. Find the maximum value of $|z^2 +kz+1|$ if $|z|=1$.
If $\sin t+\cos t=1$, and $s=\cos t +i\sin t$, compute $f(s)=1+s+s^2+\cdots +s^n$
Let $\gamma_i$ and $\overline{\gamma_i}$ be the 10 zeros of $x^{10}+(13x-1)^{10}$, where $i=1, 2, 3, 4, 5$. Compute $$\frac{1}{\gamma_1 \overline{\gamma_1}}+\frac{1}{\gamma_2 \overline{\gamma_2}}+\cdots+\frac{1}{\gamma_5 \overline{\gamma_5}}$$
If complex numbers $z_1, z_2, z_3$ satisfy
$$
\left\{
\begin{array}{l}
|z_1|=|z_2|=|z_3|=1\\
\\
\displaystyle\frac{z_1}{z_2}+\frac{z_2}{z_3}+\frac{z_3}{z_1}=1
\end{array}
\right.
$$
Compute $|az_1 +bz_2+cz_3|$ where $a, b, c$ are three given real numbers.
Show that $$\sin\frac{\pi}{2n+1}\cdot\sin\frac{2\pi}{2n+1}\cdots\sin\frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$$
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe.
Among all pairs of real numbers $(x, y)$ such that $\sin\sin x=\sin\sin y$ with $-10\pi \le x, y \le 10\pi$. Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.
Prove that if $p$ and $(p^2 + 8)$ are prime, then $(p^3 + 8p + 2)$ is prime.
Show that if $k \ge 4$, then $lcm(1; 3;\cdots; 2k- 3; 2k- 1) > (2k + 1)^2$ where $lcm$ stands for least common multiple.
Find all positive integers $n$ such that for all odd integers $a$. If $a^2\le n$, then $a|n$.