Found problems: 32
2024 AMC 10, 10
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of triangle $CFB$?
$\textbf{(A) } 5 : 4 \qquad \textbf{(B) } 4 : 3 \qquad \textbf{(C) } 3 : 2 \qquad \textbf{(D) } 5 : 3 \qquad \textbf{(E) } 2 : 1$
2024 AMC 10, 19
In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the $12$ entries will be "Possible"?
\begin{tabular}{|c|c|c|c|c|} \cline{2-5}
\multicolumn{1}{c|}{} & \textbf{zero} & \textbf{exactly one} & \textbf{exactly two} & \textbf{more than two}\\ \hline
\textbf{zero slope} & ? & ? & ? & ?\\ \hline
\textbf{nonzero rational slope} & ? & ? & ? & ?\\ \hline
\textbf{irrational slope} & ? & ? & ? & ?\\ \hline
\end{tabular}
$
\textbf{(A) }4 \qquad
\textbf{(B) }5 \qquad
\textbf{(C) }6 \qquad
\textbf{(D) }7 \qquad
\textbf{(E) }9 \qquad
$
2024 AMC 10, 17
In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second, and Bruna is fifth. How many different results of the race are possible?
$
\textbf{(A) }180 \qquad
\textbf{(B) }361 \qquad
\textbf{(C) }420 \qquad
\textbf{(D) }431 \qquad
\textbf{(E) }720 \qquad
$
2024 AMC 10, 20
Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?
$
\textbf{(A) }60\qquad
\textbf{(B) }72\qquad
\textbf{(C) }90\qquad
\textbf{(D) }108\qquad
\textbf{(E) }120\qquad
$
2024 AMC 10, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2024 AMC 10, 16
Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase $1, 2, 3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time?
$
\textbf{(A) }1010 \qquad
\textbf{(B) }1011 \qquad
\textbf{(C) }1012 \qquad
\textbf{(D) }1013 \qquad
\textbf{(E) }1014 \qquad
$
2024 AMC 12/AHSME, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$