Olympiad problems

2017 AMC 12/AHSME, 18

Tags: AMC , AMC 10 , AMC 10 A , AMC 12 , AMC 12 A , 2017 AMC 10A , 2017 AMC 12A

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$

2017 AMC 10, 5

Tags: AMC , AMC 10 , AMC 10 A , AMC 12 A , 2017 AMC 10A , 2017 AMC 12A , AMC 12

The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$

2017 AMC 10, 8

Tags: AMC , AMC 10 , 2017 AMC 10A , counting , AMC 12 , 2017 AMC 12A

At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? $\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

2017 AMC 12/AHSME, 3

Tags: AMC , AMC 10 , AMC 12 , 2017 AMC 12A , 2017 AMC 10A

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically? $\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\ $\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\ $\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\ $\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\ $\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.

2017 AMC 10, 16

Tags: AMC , AMC 10 , AMC 10 A , 2017 AMC 10A , AMC 12 , 2017 AMC 12A

There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

2017 AMC 10, 10

Tags: 2017 AMC 10A , 2017 AMC 12A , AMC 10 , AMC 12 , counting , AMC

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

2017 AMC 10, 9

Tags: AMC , AMC 10 , AMC 10 A , 2017 AMC 10A

Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km all downhill, and then back to town $A$, a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$-km ride than it takes Penny? $\textbf{(A)}\ 45\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 65\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 95$

2017 AMC 10, 22

Tags: AMC , AMC 10 , geometry , 2017 AMC 10A

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? $ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$

2017 AMC 12/AHSME, 5

Tags: AMC , AMC 10 , 2017 AMC 10A , counting , AMC 12 , 2017 AMC 12A

At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? $\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

2017 AMC 12/AHSME, 10

Tags: AMC , AMC 10 , probability , AMC 12 , 2017 AMC 10A , 2017 AMC 12A

Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number? $\textbf{(A)}~\frac12 \qquad \textbf{(B)}~\frac23 \qquad \textbf{(C)}~\frac34 \qquad \textbf{(D)}~\frac56\qquad \textbf{(E)}~\frac78$

2017 AMC 12/AHSME, 12

Tags: AMC , AMC 10 , AMC 10 A , 2017 AMC 10A , AMC 12 , 2017 AMC 12A

There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

2017 AMC 10, 15

Tags: AMC , AMC 10 , probability , AMC 12 , 2017 AMC 10A , 2017 AMC 12A

Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number? $\textbf{(A)}~\frac12 \qquad \textbf{(B)}~\frac23 \qquad \textbf{(C)}~\frac34 \qquad \textbf{(D)}~\frac56\qquad \textbf{(E)}~\frac78$

2017 AMC 12/AHSME, 23

Tags: AMC , AMC 10 , AMC 12 , algebra , polynomial , 2017 AMC 10A , 2017 AMC 12A

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$? $\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$