Find all positive integer $n$ such that $(3^{2n+1} -2^{2n+1}- 6^n)$ is a composite number.
Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, find the probability that all three mathematics textbooks end up in the same box.
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person.
Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Find the $1000^{th}$ number in $S$ (in base $10$).
Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a < b < c < d \le 10$, and $a+d>b+c$. How many interesting ordered quadruples are there?
Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + e$.
In the diagram, $AB$ is the diameter of the semicircle, $\angle{CAB} = 45^\circ$, $E$ is the midpoint of $AC$, and $DE \parallel AB$. Find $\angle{ACD}$ in degrees.
Solve in integer: $36((xy+1)z+x)=475(yz+1)$
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a$.
Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$, where the $a_i$'s are integers, and $0 \le a_i \le 99$. An example of such a representation is $1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0$. Find $N$.
Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\le a < b \le 20$. Find the greatest positive integer $n$ such that $2^n$ divides $K$.
There are $n$ points, $A_1$, $A_2$, $\cdots$, $A_n$ on a line segment, $\overline{A_0A_{n+1}}$. The point $A_0$ is black, $A_{n+1}$ is white, and the rest points are colored randomly either black or white. Prove: among these $n+1$ line segments $A_kA_{k+1}$, where $k=0, 1, \cdots, n$, the number of those with different colored ending points is odd.
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$?
What is the coefficient of $x^{28}$ in the expansion of the following polynomial? \[\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,\]
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
The expression $(x+y+z)^{2006}+(x-y-z)^{2006}$ can be simplified by expanding it and combining like terms. How many terms are there in the simplified expression?
How many ordered triples of integers $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$?
$\textbf{Multiplication}$
In the multiplication problem below, $A$, $B$, $C$, and $D$ are different digits. What is $A+B$? $$\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$$
The letters $A$, $B$, $C$ and $D$ represent digits.
what digit does $D$ represent?
On a standard die with six faces, each face contains a different number from 1 through 6. Jake has a non-standard die with six faces, and each face on Jake's die contains an expression with a different value from 1 through 6. In no particular order, the six expressions are $a + 1$, $2a - 5$, $3a - 10$, $b + 8$, $2b + 5$ and $3b + 10$. If $a$ and $b$ are integers, what is the value of the product $a \times b$?
Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.