The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?
Points $C$ and $D$ are chosen on the sides of right triangle $ABE$, as shown, such that the four segments $AB$, $BC$, $CD$ and $DE$ each have length 1 inch. What is the measure of angle $BAE$, in degrees? Express your answer as a decimal to the nearest tenth.
A restaurant sells three sizes of drinks: small for \$1.20, medium for \$1.30 and large for \$1.80. Each person at a table of ten ordered one drink, for a total cost of \$14.90, before sales tax. How many people ordered a large drink?
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
Each of the 25 cells in a five-by-five grid of squares is filled with a 0, 1 or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are considered neighbors if they share a side. How many different arrangements are possible?
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
Convex quadrilateral $ABCD$ has $AB=3$, $BC=4$, $CD=13$, $AD=12$, and $\angle ABC=90^{\circ}$, as shown. What is the area of the quadrilateral?
Fifty tickets numbered with consecutive integers are in a jar. Two are drawn at random and without replacement. What is the probability that the absolute difference between the two numbers is $10$ or less?
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a \geq{1}$ and $a+b+c \leq{7}$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$.
In regular pentagon $ABCDE$, point $M$ is the midpoint of side $AE$, and segments $AC$ and $BM$ intersect at point $Z$. If $ZA$ = 3, what is the value of $AB$? Express your answer in simplest radical form.
A set $\mathbb{S}$ consists of triangles whose sides have integer lengths less than $5$, and no two elements of $\mathbb{S}$ are congruent or similar. What is the largest number of elements that $\mathbb{S}$ can have?
Real numbers $a$ and $b$ are chosen with $1 < a < b$ such that no triangle with positive area has side lengths $1$, $a$, and $b$ or $\frac{1}{b}$, $\frac{1}{a}$, and $1$. What is the smallest possible value of $b$?
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?
When $p = \sum\limits_{k=1}^{6} k \ln{k}$, the number $e^p$ is an integer. What is the largest power of 2 that is a factor of $e^p$?
Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?
Let $P$ be the parabola with equation $y=x^2$ and let $Q = (20, 14)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r$ < $m$ < $s$. What is $r + s$?
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
How many ways are there to arrange the digits 1 through 9 in this $3 \times 3$ grid, such that the numbers are increasing from left to right in each row and increasing from top to bottom in each column?
For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$ ?
In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0 < N < 10$, it will jump to pad $(N-1)$ with probability $\frac{N}{10}$ and to pad $(N+1)$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?