Difference between revisions of "2021 April MIMC 10"

(Problem 14)
(Problem 14)
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==Problem 14==
 
==Problem 14==
 
James randomly choose an ordered pair <math>(x,y)</math> which both <math>x</math> and <math>y</math> are elements in the set  <math>\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}</math>, <math>x</math> and <math>y</math> are not necessarily distinct, and all of the equations:
 
James randomly choose an ordered pair <math>(x,y)</math> which both <math>x</math> and <math>y</math> are elements in the set  <math>\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}</math>, <math>x</math> and <math>y</math> are not necessarily distinct, and all of the equations:
<math>x+y</math>, <math>x^2+y^2</math>, <math>x^4+y^4</math>
+
<cmath>x+y</cmath>, <cmath>x^2+y^2</cmath>, <cmath>x^4+y^4</cmath>
 
are divisible by <math>5</math>. Find the probability that James can do so.
 
are divisible by <math>5</math>. Find the probability that James can do so.
  

Revision as of 17:25, 22 April 2021

Problem 1

What is the sum of $2^{3}-(-3^{4})-3^{4}+1$?

$\textbf{(A)} ~-155 \qquad\textbf{(B)} ~-153 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~9 \qquad\textbf{(E)} ~171$

Solution

Problem 2

Okestima is reading a $150$ page book. He reads a page every $\frac{2}{3}$ minutes, and he pauses $3$ minutes when he reaches the end of page 90 to take a break. He does not read at all during the break. After, he comes back with food and this slows down his reading speed. He reads one page in $2$ minutes. If he starts to read at $2:30$, when does he finish the book?

$\textbf{(A)} ~4:33 \qquad\textbf{(B)} ~5:30 \qquad\textbf{(C)} ~5:33 \qquad\textbf{(D)} ~6:30 \qquad\textbf{(E)} ~7:33$

Solution

Problem 3

Find the number of real solutions that satisfy the equation $(x^2+2x+2)^{3x+2}=1$.

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4$

Solution

Problem 4

Stiskwey wrote all the possible permutations of the letters $AABBCCCD$ ($AABBCCCD$ is different from $AABBCCDC$). How many such permutations are there?

$\textbf{(A)} ~420 \qquad\textbf{(B)} ~630 \qquad\textbf{(C)} ~840 \qquad\textbf{(D)} ~1680 \qquad\textbf{(E)} ~5040$

Solution

Problem 5

5. Given $x:y=5:3, z:w=3:2, y:z=2:1$, Find $x:w$.

$\textbf{(A)} ~3:1 \qquad\textbf{(B)} ~10:3 \qquad\textbf{(C)} ~5:1 \qquad\textbf{(D)} ~20:3 \qquad\textbf{(E)} ~10:1$

Solution

Problem 6

A worker cuts a piece of wire into two pieces. The two pieces, $A$ and $B$, enclose an equilateral triangle and a square with equal area, respectively. The ratio of the length of $B$ to the length of $A$ can be expressed as $a\sqrt[b]{c}:d$ in the simplest form. Find $a+b+c+d$.

$\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~14 \qquad\textbf{(E)} ~15$

Solution

Problem 7

Find the least integer $k$ such that $838_k=238_k+1536$ where $a_k$ denotes $a$ in base-$k$.

$\textbf{(A)} ~12 \qquad\textbf{(B)} ~13 \qquad\textbf{(C)} ~14 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16$

Solution

Problem 8

In the morning, Mr.Gavin always uses his alarm to wake him up. The alarm is special. It always rings in a cycle of ten rings. The first ring lasts $1$ second, and each ring after lasts twice the time than the previous ring. Given that Mr.Gavin has an equal probability of waking up at any time, what is the probability that Mr.Gavin wakes up and end the alarm during the tenth ring?

$\textbf{(A)} ~\frac{511}{1023} \qquad\textbf{(B)} ~\frac{1}{2} \qquad\textbf{(C)} ~\frac{512}{1023} \qquad\textbf{(D)} ~\frac{257}{512} \qquad\textbf{(E)} ~\frac{129}{256}$

Solution

Problem 9

Find the largest number in the choices that divides $11^{11}+13^2+126$.

$\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~16$

Solution

Problem 10

If $x+\frac{1}{x}=-2$ and $y=\frac{1}{x^{2}}$, find $\frac{1}{x^{4}}+\frac{1}{y^{4}}$.

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

Solution

Problem 11

How many factors of $16!$ is a perfect cube or a perfect square?

$\textbf{(A)} ~158 \qquad\textbf{(B)} ~164 \qquad\textbf{(C)} ~180 \qquad\textbf{(D)} ~1280 \qquad\textbf{(E)} ~3000$

Solution

Problem 12

Given that $x^2-\frac{1}{x^2}=2$, what is $x^{16}-\frac{1}{x^{8}}+x^{8}-\frac{1}{x^{16}}$?

$\textbf{(A)} ~1120 \qquad\textbf{(B)} ~1180 \qquad\textbf{(C)} ~3780 \qquad\textbf{(D)} ~840\sqrt{2} \qquad\textbf{(E)} ~1260\sqrt{2}$

Solution

Problem 13

Given that Giant want to put $12$ green identical balls into $3$ different boxes such that each box contains at least two balls, and that no box can contain $7$ or more balls. Find the number of ways that Giant can accomplish this.

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~6 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~19$

Solution

Problem 14

James randomly choose an ordered pair $(x,y)$ which both $x$ and $y$ are elements in the set $\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$, $x$ and $y$ are not necessarily distinct, and all of the equations: \[x+y\], \[x^2+y^2\], \[x^4+y^4\] are divisible by $5$. Find the probability that James can do so.

$\textbf{(A)} ~\frac{1}{25} \qquad\textbf{(B)} ~\frac{2}{45} \qquad\textbf{(C)} ~\frac{11}{225} \qquad\textbf{(D)} ~\frac{4}{75} \qquad\textbf{(E)} ~\frac{13}{225}$

Solution

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