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Difference between revisions of "User:Geometry285"

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

Revision as of 18:28, 11 May 2021

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

Template:G285 Mock Problems

Problem 1

What value of $x$ 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

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