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Difference between revisions of "Mock AMC 10B Problems"

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What is the difference between <math>6+7+8+9+10</math> and <math>1+2+3+4+5</math>?
 
What is the difference between <math>6+7+8+9+10</math> and <math>1+2+3+4+5</math>?
 
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 30</math>
 
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 30</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
Line 10: Line 12:
  
 
<math>\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10  \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 35</math>
 
<math>\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10  \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 35</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 2|Solution]]
  
 
===Problem 3===
 
===Problem 3===
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<math>\textbf{(A) }(\sqrt[3]{3})^{2018} \qquad \textbf{(B) }(\sqrt{3})^{2019} \qquad \textbf{(C) }(3+\sqrt{2})^2 \qquad \textbf{(D) }(2\pi)^2 \qquad \textbf{(E) }(3+\sqrt{2})(3-\sqrt{2}) \qquad</math>
 
<math>\textbf{(A) }(\sqrt[3]{3})^{2018} \qquad \textbf{(B) }(\sqrt{3})^{2019} \qquad \textbf{(C) }(3+\sqrt{2})^2 \qquad \textbf{(D) }(2\pi)^2 \qquad \textbf{(E) }(3+\sqrt{2})(3-\sqrt{2}) \qquad</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 3|Solution]]
  
 
===Problem 4===
 
===Problem 4===
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<math>\textbf{(A) }26 \qquad \textbf{(B) }27 \qquad \textbf{(C) }28 \qquad \textbf{(D) }29 \qquad \textbf{(E) }30 \qquad</math>
 
<math>\textbf{(A) }26 \qquad \textbf{(B) }27 \qquad \textbf{(C) }28 \qquad \textbf{(D) }29 \qquad \textbf{(E) }30 \qquad</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 4|Solution]]
  
 
===Problem 5===
 
===Problem 5===
Line 30: Line 38:
  
 
<math>\textbf{(A)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{4}\qquad\textbf{(D)}\ \sqrt{3}\qquad\textbf{(E)}\ 2</math>
 
<math>\textbf{(A)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{4}\qquad\textbf{(D)}\ \sqrt{3}\qquad\textbf{(E)}\ 2</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
Line 36: Line 46:
  
 
<math>\textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{1}{7}\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{2}{4}\qquad\textbf{(E)}\ \frac{2}{5}</math>
 
<math>\textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{1}{7}\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{2}{4}\qquad\textbf{(E)}\ \frac{2}{5}</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 6|Solution]]
  
 
===Problem 7===
 
===Problem 7===
Line 42: Line 54:
  
 
<math>\textbf{(A) }8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) }12 </math>
 
<math>\textbf{(A) }8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) }12 </math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 7|Solution]]
  
 
===Problem 8===
 
===Problem 8===
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<math>\mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 12</math>
 
<math>\mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 12</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 8|Solution]]
  
 
===Problem 9===
 
===Problem 9===
Line 55: Line 71:
  
 
<math>\textbf{(A)}\ 77\qquad\textbf{(B)}\ 78\qquad\textbf{(C)}\ 79\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 81</math>
 
<math>\textbf{(A)}\ 77\qquad\textbf{(B)}\ 78\qquad\textbf{(C)}\ 79\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 81</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 9|Solution]]
  
 
===Problem 10===
 
===Problem 10===
Line 61: Line 79:
  
 
<math>\mathrm{(A) \ } 100.5\qquad \mathrm{(B) \ } 1000.5\qquad \mathrm{(C) \ } 10,000.5\qquad \mathrm{(D) \ } 100,000.5\qquad \mathrm{(E) \ } 1,000,000.5</math>
 
<math>\mathrm{(A) \ } 100.5\qquad \mathrm{(B) \ } 1000.5\qquad \mathrm{(C) \ } 10,000.5\qquad \mathrm{(D) \ } 100,000.5\qquad \mathrm{(E) \ } 1,000,000.5</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 10|Solution]]
  
 
===Problem 11===
 
===Problem 11===
Line 69: Line 89:
  
 
<math>\textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50</math>
 
<math>\textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 11|Solution]]
  
 
===Problem 12===
 
===Problem 12===
Line 77: Line 99:
  
 
<math>\textbf{(A)}\ 130 \qquad\textbf{(B)}\ 131 \qquad\textbf{(C)}\ 132 \qquad\textbf{(D)}\ 133 \qquad\textbf{(E)}\ 134</math>
 
<math>\textbf{(A)}\ 130 \qquad\textbf{(B)}\ 131 \qquad\textbf{(C)}\ 132 \qquad\textbf{(D)}\ 133 \qquad\textbf{(E)}\ 134</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 12|Solution]]
  
 
===Problem 13===
 
===Problem 13===
Line 83: Line 107:
  
 
<math>\mathrm{(A) \ } \frac{1}{30}\qquad \mathrm{(B) \ } \frac{1}{31}\qquad \mathrm{(C) \ } \frac{1}{32}\qquad \mathrm{(D) \ } \frac{1}{33}\qquad \mathrm{(E) \ } \frac{1}{34}</math>
 
<math>\mathrm{(A) \ } \frac{1}{30}\qquad \mathrm{(B) \ } \frac{1}{31}\qquad \mathrm{(C) \ } \frac{1}{32}\qquad \mathrm{(D) \ } \frac{1}{33}\qquad \mathrm{(E) \ } \frac{1}{34}</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 13|Solution]]
  
 
===Problem 14===
 
===Problem 14===
Line 89: Line 115:
  
 
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 33</math>
 
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 33</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 14|Solution]]
  
 
===Problem 15===
 
===Problem 15===
Line 96: Line 124:
  
 
<math>\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15</math>
 
<math>\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 15|Solution]]
  
 
===Problem 16===
 
===Problem 16===
Line 102: Line 132:
  
 
<math>\textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 15</math>
 
<math>\textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 15</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 16|Solution]]
  
 
===Problem 17===
 
===Problem 17===
Line 110: Line 142:
  
 
<math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 35 \qquad\textbf{(E)}\ 70</math>
 
<math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 35 \qquad\textbf{(E)}\ 70</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 17|Solution]]
  
 
===Problem 18===
 
===Problem 18===
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<math>\textbf{(A) } 250\pi \qquad \textbf{(B) } 260\pi \qquad \textbf{(C) } 270\pi \qquad \textbf{(D) } 280\pi \qquad \textbf{(E) } 290\pi</math>
 
<math>\textbf{(A) } 250\pi \qquad \textbf{(B) } 260\pi \qquad \textbf{(C) } 270\pi \qquad \textbf{(D) } 280\pi \qquad \textbf{(E) } 290\pi</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 18|Solution]]
  
 
===Problem 19===
 
===Problem 19===
Line 124: Line 160:
  
 
<math>\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 64 \qquad\textbf{(D)}\ 128 \qquad\textbf{(E)}\ 256</math>
 
<math>\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 64 \qquad\textbf{(D)}\ 128 \qquad\textbf{(E)}\ 256</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 19|Solution]]
  
 
===Problem 20===
 
===Problem 20===
Line 130: Line 168:
  
 
<math>\textbf{(A) }36 \qquad \textbf{(B) }48 \qquad \textbf{(C) }56 \qquad \textbf{(D) }64 \qquad \textbf{(E) }72 \qquad</math>
 
<math>\textbf{(A) }36 \qquad \textbf{(B) }48 \qquad \textbf{(C) }56 \qquad \textbf{(D) }64 \qquad \textbf{(E) }72 \qquad</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 20|Solution]]
  
 
===Problem 21===
 
===Problem 21===
Line 140: Line 180:
  
 
<math>\textbf{(A)}\ 432 \qquad\textbf{(B)}\ 576 \qquad\textbf{(C)}\ 864 \qquad\textbf{(D)}\ 1,152 \qquad\textbf{(E)}\ 1,296</math>
 
<math>\textbf{(A)}\ 432 \qquad\textbf{(B)}\ 576 \qquad\textbf{(C)}\ 864 \qquad\textbf{(D)}\ 1,152 \qquad\textbf{(E)}\ 1,296</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 21|Solution]]
  
 
===Problem 22===
 
===Problem 22===
Line 146: Line 188:
  
 
<math>\textbf{(A)}\ 40\qquad\textbf{(B)}\ 41\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 43\qquad\textbf{(E)}\ 44</math>
 
<math>\textbf{(A)}\ 40\qquad\textbf{(B)}\ 41\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 43\qquad\textbf{(E)}\ 44</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 22|Solution]]
  
 
===Problem 23===
 
===Problem 23===
Line 152: Line 196:
  
 
<math>\textbf{(A)}\ 95\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 97\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 99</math>
 
<math>\textbf{(A)}\ 95\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 97\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 99</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 23|Solution]]
  
 
===Problem 24===
 
===Problem 24===
Line 158: Line 204:
  
 
<math>\textbf{(A)}\ 804\qquad\textbf{(B)}\ 876\qquad\textbf{(C)}\ 948\qquad\textbf{(D)}\ 984 \qquad\textbf{(E)}\ 1,020 </math>
 
<math>\textbf{(A)}\ 804\qquad\textbf{(B)}\ 876\qquad\textbf{(C)}\ 948\qquad\textbf{(D)}\ 984 \qquad\textbf{(E)}\ 1,020 </math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 24|Solution]]
  
 
===Problem 25===
 
===Problem 25===
Line 165: Line 213:
  
 
<math>\textbf{(A)}\ 374 \qquad\textbf{(B)}\ 375 \qquad\textbf{(C)}\ 503 \qquad\textbf{(D)}\ 750 \qquad\textbf{(E)}\ 751</math>
 
<math>\textbf{(A)}\ 374 \qquad\textbf{(B)}\ 375 \qquad\textbf{(C)}\ 503 \qquad\textbf{(D)}\ 750 \qquad\textbf{(E)}\ 751</math>
 +
 +
[[2019 Mock AMC 10B  Problems/Problem 25|Solution]]

Revision as of 17:06, 22 September 2019

Problem 1

What is the difference between $6+7+8+9+10$ and $1+2+3+4+5$? $\textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 30$

Solution

Problem 2

Al, Bob, Clayton, Derek, Ethan, and Frank are six Boy Scouts that will be split up into two groups of three Boy Scouts for a boating trip. How many ways are there to split up the six boys if the two groups are indistinguishable?


$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 35$

Solution

Problem 3

Which of these numbers is a rational number?


$\textbf{(A) }(\sqrt[3]{3})^{2018} \qquad \textbf{(B) }(\sqrt{3})^{2019} \qquad \textbf{(C) }(3+\sqrt{2})^2 \qquad \textbf{(D) }(2\pi)^2 \qquad \textbf{(E) }(3+\sqrt{2})(3-\sqrt{2}) \qquad$

Solution

Problem 4

In the diagram below, $ABC$ is an isosceles right triangle with a right angle at $B$ and with a hypotenuse of $40\sqrt{2}$ units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside $ABC$.

[asy] label("$B$", (8.5, -0.5), S); label("$A$", (8.5, (9sqrt(2)+0.5)), S); label("$C$", ((9.5+9sqrt(2)), -0.5), S); draw((9,0)--((9+9sqrt(2)),0)); draw((9,0)--(9,9sqrt(2))); draw(((9+9sqrt(2)),0)--(9,9sqrt(2))); draw(arc((9,0),9,0,90)); [/asy]

$\textbf{(A) }26 \qquad \textbf{(B) }27 \qquad \textbf{(C) }28 \qquad \textbf{(D) }29 \qquad \textbf{(E) }30 \qquad$

Solution

Problem 5

The three medians of the unit equilateral triangle $ABC$ intersect at point $P$. Find $PA + PB + PC$.

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

Solution

Problem 6

Mark rolled two standard dice. Given that he rolled two distinct values, find the probability that he rolled two primes.

$\textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{1}{7}\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{2}{4}\qquad\textbf{(E)}\ \frac{2}{5}$

Solution

Problem 7

What is the sum of the solutions to $n^2=x^2-8x+96$?, where $n$ is a positive integer?

$\textbf{(A) }8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) }12$

Solution

Problem 8

In the following diagram, Bob starts at the origin and makes a certain number of moves. A move is defined as him starting at $(x,y)$ and moves to $(x,y+1)$, $(x+1,y)$, $(x,y-1)$, and $(x-1,y)$ with equal probability. The probability that Bob will eventually reach the point $(4,3)$ is $N$. Find the number of distinct points, including $(4, 3)$, that satisfy that the probability that he will eventually reach that point is $N$.

$\mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 12$

Solution

Problem 9

Consider the line segment $OA_0$, which has two endpoints $O = (0, 0)$ and $A = (5, 0)$. $OA_n$ is constructed by rotating $OA_0$ about the point $O$ clockwise $\frac{360n}{\mu}$ degrees, where $\mu$ is a positive integer greater than 2 and $n < \mu$. After this operation, the line segments $A_0A_1$, $A_1A_2$, $A_2A_3$, $...$, $A_{n-2}A_{n-1}$, $A_{n-1}A_0$ are drawn. Let $S$ be the sum of the areas of the Triangles $OA_0A_1, OA_1A_2, OA_2A_3, ..., OA_{n-2}A_{n-1}, OA_{n-1}A_0$. As $n$ approaches infinity, $S$ approaches a constant $p$. Find $\lfloor p \rfloor$.


$\textbf{(A)}\ 77\qquad\textbf{(B)}\ 78\qquad\textbf{(C)}\ 79\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 81$

Solution

Problem 10

A certain period of time $P$ starts at exactly 6:09PM on a Tuesday and ends at exactly 6:09AM on a Thursday. Which of these numbers listed in the choices here is a possible length in days for $P$?

$\mathrm{(A) \ } 100.5\qquad \mathrm{(B) \ } 1000.5\qquad \mathrm{(C) \ } 10,000.5\qquad \mathrm{(D) \ } 100,000.5\qquad \mathrm{(E) \ } 1,000,000.5$

Solution

Problem 11

Consider Square $ABCD$, a square with side length $10$. Let Points $E$, $F$, $G$, $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. Find the area of the square formed by the four line segments $AG$, $BH$, $CE$, and $DF$.

[asy] draw((0,0)--(10,0)); draw((0,0)--(0, 10)); draw((10,0)--(10, 10)); draw((10,10)--(0, 10)); draw((0,10)--(5, 0)); draw((0,0)--(10, 5)); draw((10,0)--(5, 10)); draw((10,10)--(0, 5)); [/asy]

$\textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50$

Solution

Problem 12

[asy] draw((0,0)--(10*sqrt(3),0)); draw((0,0)--(0, 10)); draw((10*sqrt(3),0)--(0, 10)); draw(arc((5*sqrt(3),0),5*sqrt(3),0,180)); label("$A$",(3,4)); [/asy]

In the figure shown here, the triangle has two legs of length $10$ and $10\sqrt{3}$, and the semicircle has diameter $10\sqrt{3}$. The area of Region $A$ can be expressed as $\frac{a\pi+b\sqrt{c}}{d}$, where $a, b, c, d$ are positive integers, $c$ is square-free, $\text{ gcd }(a, d) = 1$, and $\text{ gcd }(b, d) = 1$. Find $a+b+c+d$.

$\textbf{(A)}\ 130 \qquad\textbf{(B)}\ 131 \qquad\textbf{(C)}\ 132 \qquad\textbf{(D)}\ 133 \qquad\textbf{(E)}\ 134$

Solution

Problem 13

Kevin has a girlfriend named Anna. The two of them are both in the same class, BC Calculus, which is a class that has $32$ students. To split the class up into partners that work on a group project involving integrals, the teacher, Mrs. Jannesen, randomly partitions the class into groups of two. If is assigned to be partners with his girlfriend, he will be happy. What is the probability that Kevin is happy?

$\mathrm{(A) \ } \frac{1}{30}\qquad \mathrm{(B) \ } \frac{1}{31}\qquad \mathrm{(C) \ } \frac{1}{32}\qquad \mathrm{(D) \ } \frac{1}{33}\qquad \mathrm{(E) \ } \frac{1}{34}$

Solution

Problem 14

Let $S$ be the number of distinct triangles that can be formed from $5$ coplanar points. Find the sum of all possible values of $S$.

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 33$

Solution

Problem 15

In the figure below, a square of area $108$ is inscribed inside a square of area $144$. There are two segments, labeled $m$ and $n$. The value of $m$ can be expressed as $a + b \sqrt{c}$, where $a, b, c$ are positive integers and $c$ is square-free. Find $a+b+c$.

[asy] draw((0,2)--(2,2)--(2,0)--(0,0)--cycle); draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle); label("$n$",(-0.1,0.15)); label("$m$",(-0.1,1.15));[/asy]

$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$

Solution

Problem 16

For a particular positive integer $n$, the number of ordered sextuples of positive integers $(a, b, c, d, e, f)$ that satisfy $a+b+c+d+e+f \leq n$ is exactly $3003$. Find $n$.

$\textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 15$

Solution

Problem 17

Let $S$ be a regular octagon. How many distinct quadrilaterals can be formed from the vertices of $S$ given that two quadrilaterals are not distinct if the latter can be obtained by a rotation of the former?

[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy]

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 35 \qquad\textbf{(E)}\ 70$

Solution

Problem 18

Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. What is the volume of the set of points swept out by the larger log as it rolls over the smaller one?

[asy] draw(Circle((0,0),1)); draw(Circle((-2*sqrt(3),2),3)); [/asy]

$\textbf{(A) } 250\pi \qquad \textbf{(B) } 260\pi \qquad \textbf{(C) } 270\pi \qquad \textbf{(D) } 280\pi \qquad \textbf{(E) } 290\pi$

Solution

Problem 19

What is the largest power of $2$ that divides $3^{2016}-1$?

$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 64 \qquad\textbf{(D)}\ 128 \qquad\textbf{(E)}\ 256$

Solution

Problem 20

Define a permutation $a_1a_2a_3a_4a_5a_6$ of the set $1, 2, 3, 4, 5, 6$ to be $\text{ factor-hating }$ if $\text{ gcd }(a_k, a_{k+1}) = 1$ for all $1 \leq k \leq 5$. Find the number of $\text{ factor-hating }$ permutations.

$\textbf{(A) }36 \qquad \textbf{(B) }48 \qquad \textbf{(C) }56 \qquad \textbf{(D) }64 \qquad \textbf{(E) }72 \qquad$

Solution

Problem 21

There are $N$ distinct $4\times4$ arrays of integers that satisfy: 1. Each integer in the array is a $1, 2, 3$ or $4$. 2. Every row and column contains all the integers $1, 2, 3$ and $4$. 3. No row or column contains two of the same number. Find $N$.

$\textbf{(A)}\ 432 \qquad\textbf{(B)}\ 576 \qquad\textbf{(C)}\ 864 \qquad\textbf{(D)}\ 1,152 \qquad\textbf{(E)}\ 1,296$

Solution

Problem 22

Let $S = \{r_1, r_2, r_3, ..., r_{\mu}\}$ be the set of all possible remainders when $15^{n} - 7^{n}$ is divided by $256$, where $n$ is a positive integer and $\mu$ is the number of elements in $S$. The sum $r_1 + r_2 + r_3 + ... + r_{\mu}$ can be expressed as \[p^qr,\]where $p, q, r$ are positive integers and $p$ and $r$ are as small as possible. Find $p+q+r$.

$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 41\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 43\qquad\textbf{(E)}\ 44$

Solution

Problem 23

Four real numbers $x_1, x_2, x_3, x_4$ are randomly and independently selected from the range $[0, 9]$. Let the Sets $S_1$, $S_2$, $S_3$, $S_4$ contain all of the real numbers in the range $[x_1, x_1+1], [x_2, x_2+1], [x_3, x_3+1],$ and $[x_4, x_4+1]$, respectively. The probability that the four aforementioned sets are disjoint can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

$\textbf{(A)}\ 95\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 97\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 99$

Solution

Problem 24

At a political discussion meeting held by Stalin, FDR, and Hitler, four communists (Stalin's team), four capitalists (FDR's team), and four fascists (Hitler's team) sit around a round table with $12$ seats. To encourage political debate, there is a rule that no two people of the same political stance may sit adjacent to each other. Let $N$ be the number of distinct seating arrangements following the rule. Find $\frac{N}{(4!)^3}$.

$\textbf{(A)}\ 804\qquad\textbf{(B)}\ 876\qquad\textbf{(C)}\ 948\qquad\textbf{(D)}\ 984 \qquad\textbf{(E)}\ 1,020$

Solution

Problem 25

Let $S_{n, k} = \sum_{a=0}^{n} \dbinom{a}{k}\dbinom{n-a}{k}$. Find the remainder when $\sum_{n=0}^{200} \sum_{k=0}^{200} S_{n, k}$ is divided by $1000$.


$\textbf{(A)}\ 374 \qquad\textbf{(B)}\ 375 \qquad\textbf{(C)}\ 503 \qquad\textbf{(D)}\ 750 \qquad\textbf{(E)}\ 751$

Solution

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