This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 23

2016-2017 SDML (Middle School), 7
Tags: SDML Middle School , 2016-17 SDML Round 2
Point $P$ is selected at random from the interior of the pentagon with vertices $A = (0, 2), B = (4, 0), C = (2\pi + 1, 0), D = (2\pi + 1, 4),$ and $E = (0, 4)$. What is the probability that $\angle ABP$ is obtuse? Express your answer as a common fraction.

2016-2017 SDML (Middle School), 2
Tags: SDML Middle School , 2016-17 SDML Round 2 , analytic geometry
On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square?

2016-2017 SDML (Middle School), 4
Tags: SDML Middle School , 2016-17 SDML Round 2
What is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }8\qquad\text{(D) }9\qquad\text{(E) }11$

2016-2017 SDML (Middle School), 11
Tags: SDML Middle School , 2016-17 SDML Round 2
Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any previous boxes. How many balls in total are in the boxes as a result of Emily's $2017$th step? $\text{(A) }9\qquad\text{(B) }11\qquad\text{(C) }13\qquad\text{(D) }15\qquad\text{(E) }17$

2016-2017 SDML (Middle School), 2
Tags: SDML Middle School , 2016-17 SDML Round 2
Each term of the sequence $5, 12, 19, 26, \cdots$ is $7$ more than the term that precedes it. What is the first term of the sequence that is greater than $2017$? $\text{(A) }2018\qquad\text{(B) }2019\qquad\text{(C) }2020\qquad\text{(D) }2021\qquad\text{(E) }2022$

2016-2017 SDML (Middle School), 15
Tags: SDML Middle School , 2016-17 SDML Round 2
A regular hexagon $ABCDEF$ has area $36$. Find the area of the region which lies in the overlap of the triangles $ACE$ and $BDF$. $\text{(A) }3\qquad\text{(B) }9\qquad\text{(C) }12\qquad\text{(D) }18\qquad\text{(E) }24$

2016-2017 SDML (Middle School), 10
Tags: inequalities , SDML Middle School , 2016-17 SDML Round 2
For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$ $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

2016-2017 SDML (Middle School), 1
Tags: SDML Middle School , 2016-17 SDML Round 2
What is the integer value of $\left(\sqrt{3}^{\sqrt{2}}\right)^{\sqrt{8}}$?

2016-2017 SDML (Middle School), 7
Tags: SDML Middle School , 2016-17 SDML Round 2
If $f(1) = 1$ and $f(n+1) = \frac{2f(n) + 1}{2}$, then find $f(237)$. $\text{(A) }117\qquad\text{(B) }118\qquad\text{(C) }119\qquad\text{(D) }120\qquad\text{(E) }121$

2016-2017 SDML (Middle School), 6
Tags: SDML Middle School , 2016-17 SDML Round 2
There are $4$ pairs of men and women, and all $8$ people are arranged in a row so that in each pair the woman is somewhere to the left of the man. How many such arrangements are there?

2016-2017 SDML (Middle School), 1
Tags: SDML Middle School , 2016-17 SDML Round 2 , NEEDS DIAGRAM
A "domino" is made up of two small squares: [asy] unitsize(10); draw((0,0) -- (2,0) -- (2,1) -- (0,1) -- cycle); fill((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle); [/asy] Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes? [diagram requires in-line asy]

2016-2017 SDML (Middle School), 12
Tags: SDML Middle School , 2016-17 SDML Round 2
What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x-3$? $\text{(A) }6\pi\qquad\text{(B) }7\pi\qquad\text{(C) }8\pi\qquad\text{(D) }9\pi\qquad\text{(E) }10\pi$

2016-2017 SDML (Middle School), 3
Tags: SDML Middle School , 2016-17 SDML Round 2
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? $\text{(A) }6000\qquad\text{(B) }7500\qquad\text{(C) }24,000\qquad\text{(D) }30,000\qquad\text{(E) }37,500$

2016-2017 SDML (Middle School), 8
Tags: SDML Middle School , 2016-17 SDML Round 2
Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$.

2016-2017 SDML (Middle School), 8
Tags: geometry , 3D geometry , SDML Middle School , 2016-17 SDML Round 2
An ice cream cone has radius $1$ and height $4$ inches. What is the number of inches in the radius of a sphere of ice cream which has the same volume of the cone? $\text{(A) }\frac{1}{2}\qquad\text{(B) }1\qquad\text{(C) }\frac{3}{2}\qquad\text{(D) }2\qquad\text{(E) }\frac{5}{2}$

2016-2017 SDML (Middle School), 6
Tags: SDML Middle School , 2016-17 SDML Round 2
What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other? $\text{(A) }\frac{1}{10}\qquad\text{(B) }\frac{2}{15}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{3}{10}\qquad\text{(E) }\frac{2}{5}$

2016-2017 SDML (Middle School), 14
Tags: SDML Middle School , 2016-17 SDML Round 2
Evaluate the sum $$\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots$$ $\text{(A) }\frac{5}{9}\qquad\text{(B) }\frac{5}{8}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{3}{4}\qquad\text{(E) }\frac{7}{9}$

2016-2017 SDML (Middle School), 13
Tags: SDML Middle School , 2016-17 SDML Round 2
If Scott rolls four fair six-sided dice, what is the probability that he rolls more 2's than 1's? $\text{(A) }\frac{8}{27}\qquad\text{(B) }\frac{25}{81}\qquad\text{(C) }\frac{103}{324}\qquad\text{(D) }\frac{421}{1296}\qquad\text{(E) }\frac{65}{162}$

2016-2017 SDML (Middle School), 5
Tags: SDML Middle School , 2016-17 SDML Round 2
A group of $25$ friends were discussing a large positive integer. "It can be divided by $1$," said the first friend. "It can be divided by $2$," said the second friend. "And by $3$," said the third friend. "And by $4$," added the fourth friend. This continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?

2016-2017 SDML (Middle School), 4
Tags: SDML Middle School , 2016-17 SDML Round 2
In a certain regular polygon, the measure of each interior angle is twice the measure of each exterior angle. How many sides does this regular polygon have?

2016-2017 SDML (Middle School), 5
Tags: SDML Middle School , 2016-17 SDML Round 2
What is the measure in degrees of the acute angle formed by the hands of a $12$-hour clock at $3:20$ PM? $\text{(A) }18\qquad\text{(B) }20\qquad\text{(C) }22\qquad\text{(D) }25\qquad\text{(E) }30$

2016-2017 SDML (Middle School), 3
Tags: SDML Middle School , 2016-17 SDML Round 2
A company that sells keychains has to pay $\mathdollar500$ in maintenance fees each day and then it pays each work $\mathdollar15$ an hour. Each worker makes $5$ keychains per hour, which are sold at $\mathdollar3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an $8$-hour workday?

2016-2017 SDML (Middle School), 9
Tags: SDML Middle School , 2016-17 SDML Round 2
Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$