Practice (EndingDigits,TheDivideByNineMethod,MODBasic,EulerFermatTheorem,CRT,TheModMethod,SquareNumber)

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455

For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?


456

In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?



457

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? 



458

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?


459
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 $a^2+b^2+c^2?$.

460
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.

462
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?

463
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 $b$?

464
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?

465
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 $e^p$?

466
Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?

467
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 $r + s$?

469

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?



470
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?

471
For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$ ?

472

In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? 



473

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?


474
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?

475
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 $m+n$ ?

476
Find the sum of all the positive solutions of $2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1$

477

Square $ABCD$ has side length $10$. Point $E$ is on $\overline{BC}$, and the area of $\bigtriangleup ABE$ is $40$. What is $BE$?


478
A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

479
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

480
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\]

481
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?