Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.
Find the sum of all the coefficients of $(x_1+x_2+ \cdots+x_{2016})^{2016}$.
Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?
A highly valued secretive recipe is kept in a safe with multiple locks. Keys to these locks are distributed among 9 members of the managing committee. The established rule requires that at least 6 members must present in order to open the safe. What is the minimal number of locks required? Correspondingly, how many keys are required?
Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.
Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $\overline{AB}$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI=2$ and $LD=3$, then $IC= \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
A strictly increasing sequence of positive integers $a_1$, $a_2$, $a_3$, $\cdots$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ is geometric and the subsequence $a_{2k}$, $a_{2k+1}$, $a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$. Find $a_1$.
For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\displaystyle\sum_{i=0}^{50} a_ix^{i}$. Find the value of $\displaystyle\sum_{i=0}^{50} |a_i|$.
Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$
Let $n$ be a positive integer. Show that $$\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{3^2}\Big)\cdots\Big(1+\frac{1}{3^n}\Big) < 2$$
Let $x$ be a real number, show that if the value of $\left(x+\frac{1}{x}\right)$ is an integer, then the value of $\left(x^n+\frac{1}{x^n}\right)$ is an integer too.
Prove that for every positive integer $n$, there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
Find $a$ and $b$ so that $(x-1)^2$ divides $ax^4 + bx^3+1$.
Find all pairs of real numbers $a, b$, such that the polynomial $$p(x)=(a+b)x^5 + abx^2 +1$$ is divisible by $x^2 - 3x+2$.
Let $n$ be a positive integer, and for $1\le k\le n$, let $S_k$ be the sum of the products of $1, \frac{1}{2}, \cdots, \frac{1}{n}$, taken $k$ a time ($k^{th}$ elementary symmetric polynomial). Find $S_1 + S_2 + \cdots +S_n$.
A sequence satisfies $a_1 = 3, a_2 = 5$, and $a_{n+2} = a_{n+1} - a_n$ for $n \ge 1$. What is the value of $a_{2018}$?
Show that $1\cdot 1! + 2\cdot 2! + \cdots + n\cdot n! = (n+1)!-1$
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
Determine all roots, real or complex, of the following system
\begin{align}
x+y+z &= 3\\
x^2+y^2+z^2 &= 3\\
x^3+y^3+z^3 &= 3
\end{align}
The sum of two positive integers is $2310$. Show that their product is not divisible by $2310$.
Is it possible for a geometric sequence to contain three distinct prime numbers?
Is it possible to construct 12 geometric sequences to contain all the prime between 1 and 100?
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that the Cauchy equation $$f(x+y)=f(x)+f(y)$$ holds for all $x, q\in\mathbb{Q}$.