Find the number of different rectangles that satisfy the following conditions:
- Its area is $2015$
- The lengths of all its sides are integers
How many integer solutions does the equation $(x+1)(y+1)=25$ have?
On the number line, consider the point $x$ that corresponds to the value 10. Consider 24 distinct integer points $y_1, y_2 \cdots y_{24}$ on the number line such that for all $k$ such that $1\le k\le 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of $$\sum_{n=1}^{24}(\mid y_n-1 \mid + \mid y_n+1\mid)$$
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price p. $A$ day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has 12 plushies, Bob has 40, and Charlie has 52 but they all spent the same amount of money: \$42. How many plushies did Alice buy on the first day?
A function $f$ has its domain equal to the set of integers $\{0, 1, ..., 11\}$, and $f(n)\ge 0$ for all such $n$, and $f$ satisfies: $f(0) = 0$, $f(6) = 1$. If $x \ge 0$, $y\ge 0$, and $x + y\le 11$, then $f(x + y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$. Find $f(2)^2 + f(10)^2$.
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 $a(2014)$.
Real numbers $x, y, z$ satisfy the following equality: $$4(x + y + z) = x^2 + y^2 + z^2$$
Let $M$ be the maximum of $xy + yz + zx$, and let $m$ be the minimum of $xy + yz + zx$. Find $M + 10m$.
Given that $x_{n+2} =\frac{20x_{n+1}}{14x_n}$, $x_0 = 25$, $x_1 = 11$, it follows that $$\sum_{n=0}^{\infty}\frac{x_{3n}}{2^n}=\frac{p}{q}$$ for some positive
integers $p, q$ with $GCD(p, q) = 1$. Find $p + q$.
$x, y, z$ are positive real numbers that satisfy $x^3+2y^3+6z^3 = 1$. Let $k$ be the maximum possible value of $2x + y + 3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n + n$.
For nonnegative integer $n$, the following are true:
$f(0) = 0$
$f(1) = 1$
$f(n) = f(n-\frac{m(m-1)}{2})-f(\frac{m(m+1)}{2} -n)$ for integer $m$ satisfying $m \ge 2$ and $\frac{m(m-1)}{2} < n \le \frac{m(m+1)}{2}$.
Find the smallest $n$ such that $f(n) = 4$.
What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?
$\textbf{Cutting Pizza}$
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?
You have three colors {red; blue; green} with which you can color the faces of a regular octahedron ($8$ triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)
How many different ways are there to cover a $1\times 10$ grid with some $1\times 1$ and $1\times 2$ pieces without overlapping?
What is the size of the largest subset $S'$ of $S=\{ 2^x 3^y 5^z : 0\le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p\mid q$.
Let $f(n)$ be the number of points of intersections of diagonals of a $n$-dimensional hypercube that is not the vertice of the cube. For example, $f(3) = 7$ because the intersection points of a cube's diagonals are at the centers of each face and the center of the cube. Find $f(5)$
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?
Triangle $ABC$ is isosceles, and $\angle{ABC}=x^\circ$. If the sum of possible measurements of $\angle{BAC}=240^\circ$, find $x^\circ$.
Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when takes a blue pill, he will loose one pound. If Neo originally wights one pound, what is the minimum number of pills he must take to make his weight 2015 pound?
Consider all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $$f(f(x)+2x+20)=15$$ Call an integer $n$ $\textit{good}$ if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$.
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$. If $IA=5$, $IB=7$, $IC=4$, $ID=9$, find the value of $\frac{AB}{CD}$.
Find $\textit{any}$ quadruple of positive integers $(a, b, c, d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.
Condier a $9\times 9$ grid of squares. Haraki fills each square in hthis grid with integer between 1 and 9, inclusive. The grid is called a $\textit{super-sudoku}$ if each of the following three conditions hold:
- Each column is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
- Each row is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
- Each $3\times 3$ sub-grid is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
How any such super-sudokus are there?
How many numbers between $1$ and $2020$ are multiples of $3$ or $4$ but not $5$?
How many positive integers, not exceeding $2019$, are relatively prime to $2019$?