Given $$P(x)=(1+x+x^2)^{100}=a_0+a_1x+\cdots+a_{200}x^{200}$$

Compute the following sums:

• $S_1=a_0+a_1+a_2+a_3 +\cdots+a_{200}$
• $S_2=a_0+a_2+a_4+a_6 +\cdots+a_{200}$.

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

Let $x_1$ and $x_2$ be the two roots of equation $x^2 − 3x + 2 = 0$. Find the following values without computing $x_1$ and $x_2$ directly.

i) $x_1^4 + x_2^4$

ii) $x_1 - x_2$

(Note: for (i) above, how many different solutions can you find?)

Find the number of non-negative integer solutions to the following equation: $$x_1+x_2+\cdots+x_5=14$$

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 $x_2, x_3, x_4 > 0$

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?

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 $16$.

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.

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 $$(5n+1)(5n+2)(5n+3)(5n+4)\equiv -1 \pmod{25}$$

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.

Find the multiplicative order of $3$ modulo $17$.