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 ways are there to express $20$ as the sum of $1$, $2$, and $5$? (All numbers must appear.)
There are $2$ white balls, $3$ red balls, and $1$ yellow ball in a jar. How many different ways are there to retrieve $3$ balls?
There are $2$ white balls, $3$ red balls, and $1$ yellow balls in a jar. How many different ways are there to retrieve $3$ balls to form a line?
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
The sides of a right triangle all have lengths that are whole numbers. The sum of the length of one leg and the hypotenuse is 49. Find the sum of all the possible lengths of the other leg.
(A) 7 (B) 49 (C) 63 (D) 71 (E) 96
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.
Let $f(x) = x^3+ax^2+bx+c$ have solutions that are distinct negative integers. If $a+b+c =2014$, \ffind $c$.
What is the last digit of $17^{17^{17^{17}}}$?
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
Find the sum of all positive integer $x$ such that $3\times 2^x = n^2-1$ for some positive integer $n$.
Find the number of pairs of integer solution $(x, y)$ that satisfies the equation $$(x-y + 2)(x-y-2) =-(x-2)(y-2)$$
Given $S = \{2, 5, 8, 11, 14, 17, 20,\cdots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1, A2,\cdots A_n\}$, s.t. $\displaystyle\sum_{i=1}^{n}\frac{1}{A_i}=1$ Find the minimum possible value of $n$.
Find the number of positive integers $n\le 2014$ such that there exists integer $x$ that satisfies the condition that $\displaystyle\frac{x + n}{x-n}$ is an odd perfect square.
Find all number sets $(a, b, c, d)$ s.t. $1 < a \le b \le c \le d, a,b,c,d \in\mathbb{N}$, and $a^2 + b + c + d,
a + b^2 + c + d, a + b + c^2 + d$ and $a + b + c + d^2$ are all square numbers. Sum the value of $d$ across all solution $set(s)$.
Evaluate $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{1368}+\sqrt{1369}}$.
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n)) + f(n) = 2n + 3$ for all nonnegative integers $n$. Find $f(2014)$.
Given that $a_n a_{n-2} - a_{n-1}^2 +a_n-na_{n-2}=-n^2+3n-1$ and $a_0=1$, $a_1=3$, find $a_{20}$.
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.
Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\cdots$, $8$, $1$, $2$, $3$, $\cdots$, $8$, $\cdots$ in order. Line segments may only be drawn to connect points labeled with the same number. What is the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).
Consider an orange and black coloring of a $20\times 14$ square grid. Let $n$ be the number of coloring such that every row and column has an even number of orange square. Evaluate $\log_2 n$.
Find the number of fractions in the following list that is in its lowest form (i.e. the denominator and the numerator are co-prime). $$\frac{1}{2014}, \frac{2}{2013}, \frac{3}{2012}, \cdots, \frac{1007}{1008}$$
For all positive integer $n$, show that $$\sum_{k=1}^n\frac{k\cdot k! \cdot\binom{n}{k}}{n^k}=n$$
Solve the equation $$\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=(a+1)\sqrt{\frac{x}{x+\sqrt{x}}}$$
There are $n$ circles inside a square $ABC$ whose side's length is $a$. If the area of any circle is no more than 1, and every line that is parallel to one side of $ABCD$ intersects at most one such circle, show that the sum of the area of all these $n$ circles is less than $a$.