Let $f(x)$ be a function defined on $\mathbb{R}$. If for every real number $x$, the relationships $$f(x+3)\le f(x)+3\quad\text{and}\quad f(x+2)\ge f(x)+2$$ always hold.
1) Show $g(x) = f(x)-x$ is a periodic function.
2) If $f(998)=1002$, compute $f(2000)$
There is a sequence with $a(2) = 0$, $a(3) = 1$ and $a(n) = a(\lfloor{\frac{n}{2}}\rfloor)+a(\lceil{\frac{n}{2}}\rceil)$ for $n\ge 4$. Find $a(2014)$.
$\textbf{Cutting Pizza}$
Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of pieces (the pizza is too heavy to be lifted up). How many finite pieces of pizza do you have?
Tom and Jerry are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2,\cdots, 2014$ are to be written. (Each number will be written exactly once). Tom fills in a piece of paper first. How many pieces of paper must Jerry fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Tom's in at least one position?
Joe has sufficient quantity of pennies, nickels, dimes, and quarters. He wants to pay a $\$1$ bill using these coins. How many different combinations does he have?
How many different weights can be measured using a set of 4 masses of 1, 2, 3, 4 grams each? For each measurable weight, how many different ways are possible?
How many different ways to express 13 as the sum of some positive odd integers? These integers do not need to be unique. Sequence of these integers also matters. For example $5 + 7 + 1$ and $7 + 1 + 5$ will be treated as two different ways.
How many different $5$-digit numbers can be formed using $1$, $2$, $3$, and $4$ that satisfy the following conditions:
- the digit $1$ must appear either $2$ or $3$ times,
- the digit $2$ must appear even times,
- the digit $3$ must appear odd times, and
- the digit $4$ has no restriction
There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends' homes.
A girl and a guy are going to arrive at a train station. If they arrive within $10$ minutes of each other, they will instantly fall in love and live happily ever after. But after $10$ minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between $8$ AM and $9$ AM with equal probability. The guy will arrive between $7$ AM and $8:30$ AM, also with equal probability. Find the probability that the probability that they fall in love.
What is the smallest positive integer $n$ such that $20\equiv n^{15} \pmod{29}$?
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3\le p<100$, and $1\le a < p$ such that the sum $a+a^2+a^3+ \cdots + a^{(p-2)!}$ is not divisible by $p$?
Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.)
- Find, with proof, the maximum possible value of $n$.
- For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard.
If for any integer $k\ne 27$ and $\big(a-k^{2015}\big)$ is divisible by $(27-k)$, what is the last two digits of $a$?
Solve $3x^{15}-x^{13}-x^{12} -x^{11} -3x^5 +6x^3 -2x^2 +2x-1\equiv 0 \pmod{11}$
Solve $$\left\{ \begin{array}{rcl} x &\equiv 2 &\pmod{3}\\ x &\equiv 2 &\pmod{5}\\ x &\equiv -3 &\pmod{7}\\x &\equiv -2 &\pmod{13} \end{array}\right.$$
Solve $$\left\{ \begin{array}{rcl} 4x & \equiv 14 &\pmod{15}\\ 9x & \equiv 11 &\pmod{20}\\ \end{array}\right.$$
Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.
Show that $x^n + 5x^{n-1} + 3 = 0$ cannot be factorized into two non-constant polynomials with integer coefficients.
A total of $2018$ tickets, numbered $1$, $2$, $3$, $\cdots$, $2014$, $2015$ are placed in an empty bag. Alfrid removes ticket $a$ from the bag. Bernice then removes ticket $b$ from the bag. Finally, Charlie removes ticket $c$ from the bag. They notice that $a < b < c$ and $a + b + c = 2018$. In how many ways could this happen?
Let $m=4k+1$ where $k$ is a non-negative integer. Show that $$a=\binom{n}{1}+m\binom{n}{3}+m^2\binom{n}{5}+\cdots+m^{\frac{n-1}{2}}\binom{n}{n}$$
is divisible by $2^{n-1}$, where $n$ is an odd number.
Let $a$, $b$ be two positive real numbers, and $n$ be a positive integer greater than $2$. Show that $$\frac{a^n+a^{n-1}b+\cdots+ab^{-1}+b^n}{n+1}\ge \Big(\frac{a+b}{2}\Big)^n$$
Show that all terms of the sequence $a_n=\left(\frac{3+\sqrt{5}}{2}\right)^n+\left(\frac{3-\sqrt{5}}{2}\right)^n -2$ are integers. And when $n$ is even, $a_n$ can be expressed as $5m^2$, when $n$ is odd $a_n$ can be expressed as $m^2$.
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
There are $n$ circles on the plane. Every pair of two circles intersect at two points. No three circles pass the same point. How many regions do these circles divide the whole plane into?