The expressions $A$ = $1\times 2 + 3\times 4 + 5\times 6 + \cdots + 37 \times 38 + 39$ and $B$ = $1 + 2 \times 3 + 4\times 5 + \cdots + 36\times 37 + 38\times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.
The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, find the probability that exactly two of the sleepers are from the same country.
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. Find the probability that Wednesday is the first day Sandy selects matching socks.
Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a cirle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.
![](data:image/png;base64,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)
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is $99$. Find the area of $FGHJ$.
![](data:image/png;base64,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)
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$.
Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n \ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.
Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.$$ Find $|f(0)|$.
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
Consider all $1000$-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. Find the arithmetic mean of all of these least elements.
With all angles measured in degrees, the product $\displaystyle\prod_{k=1}^{45} csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.
For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overset\frown{AB}$ on that face measures $120^\text{o}$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.
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)
Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.
In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. Find the probability that a randomly chosen Latin student is a sophomore.
Let $m$ be the least positive integer divisible by $17$ whose digits sum is $17$. Find $m$.
In an isosceles trapezoid, the parallel bases have lengths $log 3$ and $log 192$, and the altitude to these bases has length $log 16$. The perimeter of the trapezoid can be written in the form $log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$.
Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.
Steve says to Jon, 'I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $$P(x) = 2x^3-2ax^2+(a^2-81)x-c$$
for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?'
After some calculations, Jon says, 'There is more than one such polynomial.'
Steve says, 'You're right. Here is the value of $a$.' He writes down a positive integer and asks, 'Can you tell me the value of $c$?'
Jon says, 'There are still two possible values of $c$.' Find the sum of the two possible values of $c$.
Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = w$, the area of $PQRS$ can be expressed as the quadratic polynomial Area($PQRS$) = $\alpha w - \beta \cdot w^2$.
Then the coefficient $\beta = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b} < \frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.
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)
Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1$, $2$, $3$, $4$, $5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1$, $2$, $\ldots$, $7$.