Found problems: 35
2017 AMC 12/AHSME, 22
A square is drawn in the Cartesian coordinate plane with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x,y)$ to each of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, $(x-1,y+1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 39$
2017 AMC 10, 10
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 12/AHSME, 11
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$
2017 AMC 12/AHSME, 5
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
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
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 12/AHSME, 21
A set $S$ is constructed as follows. To begin, $S=\{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ for some $n\geq 1$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have?
$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$
2017 AMC 10, 15
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, 20
How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$?
$ \textbf{(A) \ }198\qquad \textbf{(B) \ } 199 \qquad \textbf{(C) \ } 398 \qquad \textbf{(D) \ }399\qquad \textbf{(E) \ } 597$
2017 AMC 12/AHSME, 23
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 $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$