Found problems: 5
2021 AMC 10 Fall, 7
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$
[asy]
size(6cm);
pair A = (0,10);
label("$A$", A, N);
pair B = (0,0);
label("$B$", B, S);
pair C = (10,0);
label("$C$", C, S);
pair D = (10,10);
label("$D$", D, SW);
pair EE = (15,11.8);
label("$E$", EE, N);
pair F = (3,10);
label("$F$", F, N);
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
dot(A^^B^^C^^D^^EE^^F);
draw(A--B--C--D--cycle);
draw(D--EE--F--cycle);
label("$110^\circ$", (15,9), SW);
[/asy]
$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$
2021 AMC 12/AHSME Fall, 25
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2021 AMC 10 Fall, 22
For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $?
$\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$
2021 AMC 12/AHSME Fall, 6
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$
[asy]
size(6cm);
pair A = (0,10);
label("$A$", A, N);
pair B = (0,0);
label("$B$", B, S);
pair C = (10,0);
label("$C$", C, S);
pair D = (10,10);
label("$D$", D, SW);
pair EE = (15,11.8);
label("$E$", EE, N);
pair F = (3,10);
label("$F$", F, N);
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
dot(A^^B^^C^^D^^EE^^F);
draw(A--B--C--D--cycle);
draw(D--EE--F--cycle);
label("$110^\circ$", (15,9), SW);
[/asy]
$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$
2021 AMC 12/AHSME Fall, 22
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the $3$-by-$3$ array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over?
$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 112 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\
148 \qquad\textbf{(E)}\ 160$