What is the area that is covered by putting a $8\times 6$ rectangle and a $5 \times 5$ square as shown on a table?
How many equations in the form of $ax^2+bx+c=0$ are there such that $a$, $b$, and $c$ are all single-digit prime numbers and this equation has at least one integer solution?
Restaurant MAS offers a set menu with $3$ choices of appetizers, $5$ choices of main dishes, and $2$ choices of desserts. How many possible combinations can a customer have for one appetizer, one main dish, and one dessert?
Eight chairs are arranged in two equal rows for a group of $8$. Joe and Mary must sit in the front row. Jack must sit in the back row. How many different seating plans can they have?
Two Britons, three Americans, and six Chinese form a line:
- How many different ways can the $11$ individuals line up?
- If two people of the same nationality cannot stand next to each other, how many different ways can the $11$ individuals line up?
Use digits $1$, $2$, $3$, $4$, and $5$ without repeating to create a number.
- How many 5-digit numbers can be formed?
- How many numbers will have the two even digits appearing between $1$ and $5$? (e.g.12345)
Joe plans to put a red stone, a blue stone, and a black stone on a $10 \times 10$ grid. The red stone and the blue stone cannot be in the same column. The blue stone and the black stone cannot be in the same row. How many different ways can Joe arrange these three stones?
How many different $6$-digit numbers can be formed by using digits $1$, $2$, and $3$, if no adjacent digits can be the same?
Joe wants to write $1$ to $n$ in a $1 \times n$ grid. The number 1 can be written in any grid, while the number $2$ must be written next to $1$ (can be at either side) so that these two numbers are together. The number 3 must be written next to this two-number block. This process goes on. Every new number written must stay next to the existing number block. How many different ways can Joe fill this $1 \times n$ grid?
How many positive divisors does $20$ have?
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 $p\mid q$.