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Difference between revisions of "2021 Fall MIMC 10"

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<math>\textit{This is a mock contest of the actual AMC competition. The official solution will be released on November 17th, and feel free to add more solutions beneath the official solution after that date.}</math>
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<math>\textit{This is a mock contest of the actual AMC competition.}</math>
 
==Problem 1==
 
==Problem 1==
 
What is the sum of <math>2^{3}-(-3^{4})-3^{4}+1</math>?
 
What is the sum of <math>2^{3}-(-3^{4})-3^{4}+1</math>?
  
 
<math>\textbf{(A)} ~-155 \qquad\textbf{(B)} ~-153 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~9 \qquad\textbf{(E)} ~171</math>
 
<math>\textbf{(A)} ~-155 \qquad\textbf{(B)} ~-153 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~9 \qquad\textbf{(E)} ~171</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 1|Solution]]
 
  
 
==Problem 2==
 
==Problem 2==
Line 11: Line 9:
  
 
<math>\textbf{(A)} ~4:33 \qquad\textbf{(B)} ~5:30 \qquad\textbf{(C)} ~5:33 \qquad\textbf{(D)} ~6:30 \qquad\textbf{(E)} ~7:33</math>
 
<math>\textbf{(A)} ~4:33 \qquad\textbf{(B)} ~5:30 \qquad\textbf{(C)} ~5:33 \qquad\textbf{(D)} ~6:30 \qquad\textbf{(E)} ~7:33</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 2|Solution]]
 
  
 
==Problem 3==
 
==Problem 3==
Line 19: Line 15:
  
 
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4</math>
 
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 3|Solution]]
 
  
 
==Problem 4==
 
==Problem 4==
Line 26: Line 20:
  
 
<math>\textbf{(A)} ~420 \qquad\textbf{(B)} ~630 \qquad\textbf{(C)} ~840 \qquad\textbf{(D)} ~1680 \qquad\textbf{(E)} ~5040</math>
 
<math>\textbf{(A)} ~420 \qquad\textbf{(B)} ~630 \qquad\textbf{(C)} ~840 \qquad\textbf{(D)} ~1680 \qquad\textbf{(E)} ~5040</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 4|Solution]]
 
  
 
==Problem 5==
 
==Problem 5==
Line 33: Line 25:
  
 
<math>\textbf{(A)} ~3:1 \qquad\textbf{(B)} ~10:3 \qquad\textbf{(C)} ~5:1 \qquad\textbf{(D)} ~20:3 \qquad\textbf{(E)} ~10:1</math>
 
<math>\textbf{(A)} ~3:1 \qquad\textbf{(B)} ~10:3 \qquad\textbf{(C)} ~5:1 \qquad\textbf{(D)} ~20:3 \qquad\textbf{(E)} ~10:1</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 5|Solution]]
 
  
 
==Problem 6==
 
==Problem 6==
Line 40: Line 30:
  
 
<math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~14 \qquad\textbf{(E)} ~15</math>
 
<math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~14 \qquad\textbf{(E)} ~15</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 6|Solution]]
 
  
 
==Problem 7==
 
==Problem 7==
Line 47: Line 35:
  
 
<math>\textbf{(A)} ~12 \qquad\textbf{(B)} ~13 \qquad\textbf{(C)} ~14 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16</math>
 
<math>\textbf{(A)} ~12 \qquad\textbf{(B)} ~13 \qquad\textbf{(C)} ~14 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 7|Solution]]
 
  
 
==Problem 8==
 
==Problem 8==
Line 54: Line 40:
  
 
<math>\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}</math>
 
<math>\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}</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 8|Solution]]
 
  
 
==Problem 9==
 
==Problem 9==
Line 61: Line 45:
  
 
<math>\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~16</math>
 
<math>\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~16</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 9|Solution]]
 
  
 
==Problem 10==
 
==Problem 10==
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<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>
−
 
−
[[2021 April MIMC 10 Problems/Problem 10|Solution]]
 
  
 
==Problem 11==
 
==Problem 11==
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<math>\textbf{(A)} ~158 \qquad\textbf{(B)} ~164 \qquad\textbf{(C)} ~180 \qquad\textbf{(D)} ~1280 \qquad\textbf{(E)} ~3000</math>
 
<math>\textbf{(A)} ~158 \qquad\textbf{(B)} ~164 \qquad\textbf{(C)} ~180 \qquad\textbf{(D)} ~1280 \qquad\textbf{(E)} ~3000</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 11|Solution]]
 
  
 
==Problem 12==
 
==Problem 12==
Line 82: Line 60:
  
 
<math>\textbf{(A)} ~1120 \qquad\textbf{(B)} ~1180 \qquad\textbf{(C)} ~3780 \qquad\textbf{(D)} ~840\sqrt{2} \qquad\textbf{(E)} ~1260\sqrt{2}</math>
 
<math>\textbf{(A)} ~1120 \qquad\textbf{(B)} ~1180 \qquad\textbf{(C)} ~3780 \qquad\textbf{(D)} ~840\sqrt{2} \qquad\textbf{(E)} ~1260\sqrt{2}</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 12|Solution]]
 
  
 
==Problem 13==
 
==Problem 13==
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<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~6 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~19 </math>
 
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~6 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~19 </math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 13|Solution]]
 
  
 
==Problem 14==
 
==Problem 14==
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<math>\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}</math>
 
<math>\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}</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 14 |Solution]]
 
  
 
==Problem 15==
 
==Problem 15==
Line 107: Line 79:
  
 
<math>\textbf{(A)} ~30 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~64 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126</math>
 
<math>\textbf{(A)} ~30 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~64 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 15 |Solution]]
 
  
 
==Problem 16==
 
==Problem 16==
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<math>\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~25</math>
 
<math>\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~25</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 16 |Solution]]
 
  
 
==Problem 17==
 
==Problem 17==
Line 121: Line 89:
  
 
<math>\textbf{(A)} ~4632 \qquad\textbf{(B)} ~4844 \qquad\textbf{(C)} ~4860\qquad\textbf{(D)} ~4864 \qquad\textbf{(E)} ~8960</math>
 
<math>\textbf{(A)} ~4632 \qquad\textbf{(B)} ~4844 \qquad\textbf{(C)} ~4860\qquad\textbf{(D)} ~4864 \qquad\textbf{(E)} ~8960</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 17 |Solution]]
 
  
 
==Problem 18==
 
==Problem 18==
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<math>\textbf{(E)}</math> There are two overlapping circles on the right of the <math>y</math>-axis with each area <math>4\pi</math> and the intersection area of two overlapping circles on the left of the <math>y</math>-axis with each area <math>4\pi</math>.
 
<math>\textbf{(E)}</math> There are two overlapping circles on the right of the <math>y</math>-axis with each area <math>4\pi</math> and the intersection area of two overlapping circles on the left of the <math>y</math>-axis with each area <math>4\pi</math>.
−
 
−
[[2021 April MIMC Problems/Problem 18 |Solution]]
 
  
 
==Problem 19==
 
==Problem 19==
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<math>\textbf{(A)} ~6 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~8 \qquad\textbf{(D)} ~9 \qquad  \textbf{(E)} ~\textrm{Does Not Exist}</math>
 
<math>\textbf{(A)} ~6 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~8 \qquad\textbf{(D)} ~9 \qquad  \textbf{(E)} ~\textrm{Does Not Exist}</math>
−
 
−
[[2021 April MIMC Problems/Problem 19 |Solution]]
 
  
 
==Problem 20==
 
==Problem 20==
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<math>\textbf{(A)} ~512:1 \qquad\textbf{(B)} ~1024:1 \qquad\textbf{(C)} ~2^{64}:1 \qquad\textbf{(D)} ~2^{80}:1 \qquad\textbf{(E)} ~2^{160}:1 \qquad</math>
 
<math>\textbf{(A)} ~512:1 \qquad\textbf{(B)} ~1024:1 \qquad\textbf{(C)} ~2^{64}:1 \qquad\textbf{(D)} ~2^{80}:1 \qquad\textbf{(E)} ~2^{160}:1 \qquad</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 20 |Solution]]
 
  
 
==Problem 21==
 
==Problem 21==
Line 157: Line 117:
  
 
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4 \qquad</math>
 
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4 \qquad</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 21 |Solution]]
 
  
 
==Problem 22==
 
==Problem 22==
Line 166: Line 124:
  
 
<math>\textbf{(A)} ~5 \qquad\textbf{(B)} ~15 \qquad\textbf{(C)} ~61 \qquad\textbf{(D)} ~349 \qquad\textbf{(E)} ~2009 \qquad</math>
 
<math>\textbf{(A)} ~5 \qquad\textbf{(B)} ~15 \qquad\textbf{(C)} ~61 \qquad\textbf{(D)} ~349 \qquad\textbf{(E)} ~2009 \qquad</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 22 |Solution]]
 
  
 
==Problem 23==
 
==Problem 23==
Line 182: Line 138:
  
 
<math>\textbf{(A)} ~\frac{14+10\pi}{17} \qquad\textbf{(B)} ~\frac{13+\sqrt{2}}{28} \qquad\textbf{(C)} ~\frac{4+\sqrt{2}}{7+3\pi} \qquad\textbf{(D)} ~\frac{13}{28+6\pi} \qquad\textbf{(E)} ~\frac{13}{30\pi}\qquad</math>
 
<math>\textbf{(A)} ~\frac{14+10\pi}{17} \qquad\textbf{(B)} ~\frac{13+\sqrt{2}}{28} \qquad\textbf{(C)} ~\frac{4+\sqrt{2}}{7+3\pi} \qquad\textbf{(D)} ~\frac{13}{28+6\pi} \qquad\textbf{(E)} ~\frac{13}{30\pi}\qquad</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 24 |Solution]]
 
  
 
==Problem 25==
 
==Problem 25==
Line 191: Line 145:
  
 
<math>\textbf{(A)} ~4 \qquad\textbf{(B)} ~\frac{12}{\pi} \qquad\textbf{(C)} ~\frac{2+3\sqrt{3}}{2} \qquad\textbf{(D)} ~\frac{4+\sqrt{3}}{2} \qquad\textbf{(E)} ~\frac{12+4\sqrt{2}-2\sqrt{3}}{4}\qquad</math>
 
<math>\textbf{(A)} ~4 \qquad\textbf{(B)} ~\frac{12}{\pi} \qquad\textbf{(C)} ~\frac{2+3\sqrt{3}}{2} \qquad\textbf{(D)} ~\frac{4+\sqrt{3}}{2} \qquad\textbf{(E)} ~\frac{12+4\sqrt{2}-2\sqrt{3}}{4}\qquad</math>
−
 
−
[[2021 April MIMC 10 Problems/Problem 25 |Solution]]
 
  
 
==Additional Information==
 
==Additional Information==

Latest revision as of 22:39, 12 September 2021

$\textit{This is a mock contest of the actual AMC competition.}$

Problem 1

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

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

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?

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

Problem 3

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

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

Problem 4

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

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

Problem 5

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

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

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 $a+b+c+d$.

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

Problem 7

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

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

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?

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

Problem 9

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

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

Problem 10

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

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

Problem 11

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

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

Problem 12

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

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

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.

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

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.

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

Problem 15

Paul wrote all positive integers that's less than $2021$ and wrote their base $4$ representation. He randomly choose a number out the list. Paul insist that he want to choose a number that had only $2$ and $3$ as its digits, otherwise he will be depressed and relinquishes to do homework. How many numbers can he choose so that he can finish his homework?

$\textbf{(A)} ~30 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~64 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126$

Problem 16

Find the number of permutations of $AAABBC$ such that at exactly two $A$s are adjacent, and the $B$s are not adjacent.

$\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~25$

Problem 17

The following expression \[\sum_{k=1}^{60} {60 \choose k}+\sum_{k=1}^{59} {59 \choose k}+\sum_{k=1}^{58} {58 \choose k}+\sum_{k=1}^{57} {57 \choose k}+\sum_{k=1}^{56} {56 \choose k}+\sum_{k=1}^{55} {55 \choose k}+\sum_{k=1}^{54} {54 \choose k}+...+\sum_{k=1}^{3} {3 \choose k}-2^{10}\] can be expressed as $x^{y}-z$ which both $x$ and $y$ are relatively prime positive integers. Find $2^{x}(xy+2x+z)$.

$\textbf{(A)} ~4632 \qquad\textbf{(B)} ~4844 \qquad\textbf{(C)} ~4860\qquad\textbf{(D)} ~4864 \qquad\textbf{(E)} ~8960$

Problem 18

What can be a description of the set of solutions for this: $x^{2}+y^{2}=|2x+|2y||$?

$\textbf{(A)}$ Two overlapping circles with each area $2\pi$.

$\textbf{(B)}$ Four not overlapping circles with each area $4\pi$.

$\textbf{(C)}$ There are two overlapping circles on the right of the $y$-axis with each area $2\pi$ and the intersection area of two overlapping circles on the left of the $y$-axis with each area $2\pi$.

$\textbf{(D)}$ Four overlapping circles with each area $4\pi$.

$\textbf{(E)}$ There are two overlapping circles on the right of the $y$-axis with each area $4\pi$ and the intersection area of two overlapping circles on the left of the $y$-axis with each area $4\pi$.

Problem 19

$(0.51515151...)_n$ can be expressed as $(\frac{6}{n})$ in base $10$ which $n$ is a positive integer. Find the sum of the digits of $n^{3}$.

$\textbf{(A)} ~6 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~8 \qquad\textbf{(D)} ~9 \qquad \textbf{(E)} ~\textrm{Does Not Exist}$

Problem 20

Given that $y=24\cdot 34\cdot 67\cdot 89$. Given that the product of the even divisors is $a$, and the product of the odd divisors is $b$. Find $a \colon b^4$.

$\textbf{(A)} ~512:1 \qquad\textbf{(B)} ~1024:1 \qquad\textbf{(C)} ~2^{64}:1 \qquad\textbf{(D)} ~2^{80}:1 \qquad\textbf{(E)} ~2^{160}:1 \qquad$

Problem 21

How many solutions are there for the equation $\left \lfloor{x}\right \rfloor^{2}-\left \lceil{x}\right \rceil=0$. (Recall that $\left \lfloor{x}\right \rfloor$ is the largest integer less than $x$, and $\left \lceil{x}\right \rceil$ is the smallest integer larger than $x$.)

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

Problem 22

In the diagram, $ABCD$ is a square with area $6+4\sqrt{2}$. $AC$ is a diagonal of square $ABCD$. Square $IGED$ has area $11-6\sqrt{2}$. Given that point $J$ bisects line segment $HE$, and $AE$ is a line segment. Extend $EG$ to meet diagonal $AC$ and mark the intersection point $H$. In addition, $K$ is drawn so that $JK//EC$. $FH^2$ can be represented as $\frac{a+b\sqrt{c}}{{d}}$ where $a,b,c,d$ are not necessarily distinct integers. Given that $gcd(a,b,d)=1$, and $c$ does not have a perfect square factor. Find $a+b+c+d$.

24.png

$\textbf{(A)} ~5 \qquad\textbf{(B)} ~15 \qquad\textbf{(C)} ~61 \qquad\textbf{(D)} ~349 \qquad\textbf{(E)} ~2009 \qquad$

Problem 23

On a coordinate plane, point $O$ denotes the origin which is the center of the diamond shape in the middle of the figure. Point $A$ has coordinate $(-12,12)$, and point $C$, $E$, and $G$ are formed through $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ rotation about the origin $O$, respectively. Quarter circle $BOH$ (formed by the arc $BH$ and line segments $BO$ and $GH$) has area $25\pi$. Furthermore, another quarter circle $DOF$ formed by arc $DF$ and line segments $OF$, $OD$ is formed through a reflection of sector $BOH$ across the line $y=x$. The small diamond centered at $O$ is a square, and the area of the little square is $2$. Let $x$ denote the area of the shaded region, and $y$ denote the sum of the area of the regions $ABH$ (formed by side $AB$, arc $BH$, and side $HA$), $DFE$ (formed by side $ED$, arc $DF$, and side $FE$) and sectors $FGH$ and $BCD$. Find $\frac{x}{y}$ in the simplest radical form.

19 (1).png

$\textbf{(A)} ~\frac{50\pi+1}{280} \qquad\textbf{(B)} ~\frac{50\pi\sqrt{2}+\sqrt{2}}{560} \qquad\textbf{(C)} ~\frac{50\pi+1}{140+100\pi} \qquad\textbf{(D)} ~\frac{50\pi+1}{280+100\pi} \qquad\textbf{(E)} ~\frac{50\pi^2+700\pi\sqrt{2}+3001\pi-70\sqrt{2}+60}{2\pi^2+240\pi+6920}\qquad$

Solution

Problem 24

One semicircle is constructed with diameter $AH=4$ and let the midpoint of $AH$ be $M$. Construct a point $O$ on the side of segment $AH$ (closer to segment $AH$ than arc $AH$) such that the distance from $A$ to $O$ is $2\sqrt{5}$, and that $OM$ is perpendicular to the diameter $AH$. Three more such congruent semicircles are formed through multiple $90^{\circ}$rotations around the point $O$. Name the $6$ endpoints of the diameters $B$, $C$, $D$, $E$, $F$, $G$ in a circular direction from $A$ to $H$. Another four congruent semicircles are constructed with diameters $AB, CD, EF, GH$, and that the distance from the diameters to the point $O$ are less than the distance from the arcs to the point $O$. Connect $AC$, $CD$, $DO$, $OG$, and $GA$. Find the ratio of the area of the pentagon $ACDOG$ to the total area of the shape formed by arcs $AB$, $BC$, $CD$, $DE$, $EF$, $FG$, $GH$, $HA$.

$\textbf{(A)} ~\frac{14+10\pi}{17} \qquad\textbf{(B)} ~\frac{13+\sqrt{2}}{28} \qquad\textbf{(C)} ~\frac{4+\sqrt{2}}{7+3\pi} \qquad\textbf{(D)} ~\frac{13}{28+6\pi} \qquad\textbf{(E)} ~\frac{13}{30\pi}\qquad$

Problem 25

Suppose that a researcher hosts an experiment. He tosses an equilateral triangle with area $\sqrt{3}$ $cm^2$ onto a plane that has a strip every $1$ $cm$ horizontally. Find the expected number of intersections of the strips and the sides of the equilateral triangle.

25.png

$\textbf{(A)} ~4 \qquad\textbf{(B)} ~\frac{12}{\pi} \qquad\textbf{(C)} ~\frac{2+3\sqrt{3}}{2} \qquad\textbf{(D)} ~\frac{4+\sqrt{3}}{2} \qquad\textbf{(E)} ~\frac{12+4\sqrt{2}-2\sqrt{3}}{4}\qquad$

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