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Difference between revisions of "User:Geometry285"

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==Problem 5==
 
==Problem 5==
−
Let a recursive sequence be denoted by <math>a_n</math> such that <math>a_0 = 1</math> and <math>a_1 = k</math>. Suppose <math>a_{n-1} = n+a_n</math> for <math>n>1</math>. Let an infinite arithmetic sequence <math>P</math> be such that <math>P=\{k, k+p, k+2p \cdots\}</math>. If <math>k</math> is prime, for what value of <math>p</math> will <math>k_{2021} = k-2022p+1</math>?
+
Let a recursive sequence be denoted by <math>a_n</math> such that <math>a_0 = 1</math> and <math>a_1 = k</math>. Suppose <math>a_{n-1} = n+a_n</math> for <math>n>1</math>. Let an infinite arithmetic sequence <math>P</math> be such that <math>P=\{k+1, k-p+1, k-2p+1 \cdots\}</math>. If <math>k</math> is prime, for what value of <math>p</math> will <math>k_{2021} = k-2022p+1</math>?
  
 
<math>\textbf{(A)}\ 1011\qquad\textbf{(B)}\ \frac{1011}{2}\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ \frac{2021}{2}\qquad\textbf{(E)}\ 4042</math>
 
<math>\textbf{(A)}\ 1011\qquad\textbf{(B)}\ \frac{1011}{2}\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ \frac{2021}{2}\qquad\textbf{(E)}\ 4042</math>
  
 
[[G285 MC10A Problems/Problem 5|Solution]]
 
[[G285 MC10A Problems/Problem 5|Solution]]

Revision as of 19:56, 11 May 2021

Posting here until I find a place for an upcoming mock I’m creating

Template:G285 MC10A

Problem 1

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

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

Solution

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 $k_{2021} = k-2022p+1$?

$\textbf{(A)}\ 1011\qquad\textbf{(B)}\ \frac{1011}{2}\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ \frac{2021}{2}\qquad\textbf{(E)}\ 4042$

Solution

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