Find the number of integer solutions to the following equation: $$x_1+x_2+\cdots+x_6=12$$

where $x_1, x_5\ge 0$ and $x_2, x_3, x_4 > 0$

(Hockey Sticker Identity) Show that for any positive integer $n \ge k$, the following relationship holds: $$\binom{k}{k} +\binom{k+1}{k} + \binom{k+2}{k} + \cdots + \binom{n}{k} = \binom{n+1}{k+1} $$

Find the number of integer pairs $(x, y)$ such that $x^2 + y^2 = 2019$.

If a square number's tens digit is $7$, what is its units digit?

There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.

Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?

Let $n^2$ be a square number, show that $n^2\equiv 0, 1\pmod{4}$.

Show that if $n^2$ is a square number, then $n^2\equiv 0, 1, 4, 9\pmod{16}$.

In plain English, this means that the remainder can only be $0$, $1$, $4$ or $9$ when a square number is divided by $16$.

Find the number of non-decrease sequences of length $n$ and each element is a non-negative integer not exceeding $d$.

Find, with proof, all pairs of positive integers $(n, d)$ with the following property: for every integer $S$, there exists a unique non-decreasing sequence of n integers $a_1$, $a_2$, $\cdots$, $a_n$ such that $a_1 + a_2 + \cdots + a_n = S$ and $a_n-a_1 = d$.

How many terms in this sequence are squares? $$1, 11, 111, 1111, \cdots $$

How many terms in this sequence are squares? $$4, 44, 444, 4444, \cdots$$

Let $N$ be an odd square number. Show that $N$'s tens digit must be even.

 Let $N$ be a square number. If its units digit is $6$, then its tens digit must be odd.

Let $N$ be a square number. If its tens digit is odd, then its units digit must be $6$.

Let $N$ be a square number. If its units digit is neither $4$ nor $6$, then its tens digit must be even.

A person eats $X ( > 1)$ cookies in $N$ days in the following way:

• He eats $1$ plus $1/7$ of the remaining cookies on the $1^{st}$ day 
• He eats $2$ plus $1/7$ of the remaining cookies on the $2^{nd}$ day
• $\cdots$
• Finally, he eats the last $N$ cookies on the $N^{th}$ day

What is the smallest possible value of $X$?

Let $n^2$ be a square number. Show that $n^2\equiv 0, \pm 1\pmod{5}$.

Let $n$ be any positive integer, show that $$(5n+1)(5n+2)(5n+3)(5n+4)\equiv -1 \pmod{25}$$

Compute $3^{2018} \mod{17}$.