Let $a_n=\binom{2020}{3n-1}$. Find the vale of $\displaystyle\sum_{n=1}^{673}a_n$.
Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.
Let polynomials $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ satisfy: $$P(x^5) + xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$$
Prove: $(x-1) | P(x)$
Joe puts $63$ cards, number from $1$ to $63$, on a regular chessboard in sequence. The last space on the chessboard is left empty. A card can be moved to a neighboring space if that space is empty. Joe wants to just switch the card $1$ and card $2$, but leave all other cards at their original spaces, after a series of moves. Is it possible?
Solve in integers $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} = \frac{3}{5}$
$\textbf{Lily Pads}$
There are $24$ lily pads shown below. A toad can jump from one pad to an adjacent one either horizontally or vertically, but not diagonally. Can this toad visit all the pads without stopping at a pad for more than once? It can choose any pad to start its journey.
Find all positive integer solutions to: $x^2 + 3y^2 = 1998x$.
A code consists of four different digits from $1$ to $9$, inclusive. What is the probability of selection a code that consists of four consecutive digits but not necessarily in order? Express your answer as a common fraction.
If $x$ and $y$ are positive integer solutions to the equation $x^2 - 2y^2 = 1$, then $6\mid xy$.
Prove that there exist infinite many triples of consecutive integers each of which is a sum of two squares. For example: $8 = 2^2 + 2^2$, $9 = 3^2 + 0^2$, and $10=3^1 + 1^2$
Let $x$ be a real number between 0 and 1. Find the maximum value of $x(1-x^4)$.
Find the smallest positive integer $n$ such that the last $3$ digits of $n^3$ is $888$.
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 complex number $z$ satisfy $|z|=1$. If $f(z)=|z+1+i|$ reaches its maximum and minimal values when $z=z_1$ and $z=z_2$, respectively. Compute $z_1-z_2$.
Let complex number $z$ satisfy $|z|=1$, $w = z^4-z^3-3z^2i-z+1$. Find the minimal value of $|w|$.
Let $z$ be a complex number and $k$ be a known real number. Find the maximum value of $|z^2 +kz+1|$ if $|z|=1$.
Let $\theta, a \in \mathbb{R}$ and complex number $z=(a+\cos\theta)+(2a-\sin\theta)i$. If $|z|\le 2$, find the range of $a$.
If $\sin t+\cos t=1$, and $s=\cos t +i\sin t$, compute $f(s)=1+s+s^2+\cdots +s^n$
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.
Find the smallest positive integer $n$ such that the remainder is always $1$ when $n$ is divided by $2$, $3$, $4$, $5$, or $6$. In addition, $n$ must be a multiple of $7$.
Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.
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.
Solve in positive integers $\big(1+\frac{1}{x}\big)\big(1+\frac{1}{y}\big)\big(1+\frac{1}{z}\big)=2$