Difference between revisions of "User:Geometry285"

m
m
Line 4: Line 4:
  
 
==Problem 1==
 
==Problem 1==
What value of <math>x</math> minimizes <math>|||2^|x^2| - 4|-4|-8|</math>?
+
What value of <math>x</math> minimizes <math>|||2^{|x^2|} - 4|-4|-8|</math>?
  
 
<math>\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2</math>
 
<math>\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2</math>
Line 11: Line 11:
  
 
==Problem 2==
 
==Problem 2==
Suppose Mark wanted to arrange <math>5</math> books onto a bookshelf, <math>3</math> of which are math books and <math>2</math> of which are science. If both science and math books are indistinguishable, in how many ways can Mark arrange the books on the shelf?
+
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 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)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15</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>
  
 
[[G285 MC10A Problems/Problem 2|Solution]]
 
[[G285 MC10A Problems/Problem 2|Solution]]

Revision as of 19:27, 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 $S = {1,2,3,4 \cdots n}$. 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 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

Invalid username
Login to AoPS