G285 2021 MC10A

Problem 1

What is the smallest value of $x$ that minimizes $|||2^{|x^2|} - 4|-4|-8|$ ?

$\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

Problem 2

Suppose the set $S$ denotes $S = \{1,2,3 \cdots n\}$ . Then, a subset of length $1<k<n$ is chosen. All even digits in the subset $k$ are then are put into group $k_1$ , and the odd digits are put in $k_2$ . Then, one number is selected at random from either $k_1$ or $k_2$ with equal chances. What is the probability that the number selected is a perfect square, given $n=4$ ?

$\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{11}\qquad\textbf{(C)}\ \frac{6}{11}\qquad\textbf{(D)}\ \frac{7}{13}\qquad\textbf{(E)}\ \frac{3}{5}$

Solution

Problem 3

Let $ABCD$ be a unit square. If points $E$ and $F$ are chosen on $AB$ and $CD$ respectively such that the area of $\triangle AEF = \frac{3}{2} \triangle CFE$ . What is $EF^2$ ?

$\textbf{(A)}\ \frac{13}{9}\qquad\textbf{(B)}\ \frac{8}{9}\qquad\textbf{(C)}\ \frac{37}{36}\qquad\textbf{(D)}\ \frac{5}{4}\qquad\textbf{(E)}\ \frac{13}{36}$

Solution

Problem 4

What is the smallest value of $k$ for which $2^{18k} \equiv 76 \mod 100$

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

Solution

Problem 5

Let a recursive sequence be denoted by $a_n$ such that $a_0 = 1$ and $a_1 = k$ . Suppose $a_{n-1} = n+a_n$ for $n>1$ . Let an infinite arithmetic sequence $P$ be such that $P=\{k+1, k-p+1, k-2p+1 \cdots\}$ . If $k$ is prime, for what value of $p$ will $k_{2021} = k-2022p+1$ ?