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Difference between revisions of "DMC Mock AMC 10"

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[[2024 DMC Mock 10 Problems/Problem 3|Solution]]
 
[[2024 DMC Mock 10 Problems/Problem 3|Solution]]
 +
 +
===Problem 4===
 +
 +
When <math>32, 47,</math> and <math>77</math> are divided by a positive integer <math>n</math>, the remainder is the same for all three divisions. What is the greatest possible value of <math>n</math>?
 +
 +
<math>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 30</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 4|Solution]]
 +
 +
===Problem 5===
 +
 +
Ten logicians are sitting at a table. A server comes and asks if everyone wants coffee. The first logician answers “I don’t know.” Then the second logician answers “I don’t know.” This continues, with each logician answering “I don’t know,” until the tenth logician answers “no, not everyone wants coffee.” How many of the ten logicians want coffee?
 +
 +
<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 5|Solution]]
 +
 +
===Problem 6===
 +
 +
Alice, Bob, and Charlie are sharing <math>15</math> identical candies. Because Bob is greedy, he insists that he gets at least <math>5</math> candies. Find the number of ways to distribute the candies.
 +
 +
<math>\textbf{(A)}\ 55\qquad\textbf{(B)}\ 66\qquad\textbf{(C)}\ 78\qquad\textbf{(D)}\ 91\qquad\textbf{(E)}\ 105</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 6|Solution]]
 +
 +
===Problem 7===
 +
 +
Bob is advertising the Dallas Reunion Tower by making a poster comparing its height to the Burj Khalifa. Currently, in his diagram, the image of the Burj Khalifa is five times as tall as the Reunion Tower. Bob wants to scale the image of the Reunion tower so that it is <math>90\%</math> the height of the Burj Khalifa. If the area of the image of the Reunion tower was originally <math>100</math> square inches, what is the area, in square inches, of the scaled image? (Note that scaling is done proportionately in both width and length).
 +
 +
<math>\textbf{(A)}\ 450\qquad\textbf{(B)}\ 1000\qquad\textbf{(C)}\ 2025\qquad\textbf{(D)}\ 5000\qquad\textbf{(E)}\ 8100</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 7|Solution]]
 +
 +
===Problem 8===
 +
 +
Compute <math>2020^2 + 2024^2 - 2020 \cdot 2024 + 2027^2 - 2027(2020 + 2024)</math>.
 +
 +
<math>\textbf{(A)}\ 36\qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 38\qquad\textbf{(D)}\ 39\qquad\textbf{(E)}\ 40</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 8|Solution]]
 +
 +
===Problem 9===
 +
 +
Nathan and his friends are shopping together for a birthday gift for Nathan’s new girlfriend to celebrate Nathan’s success. For each person he brings along to shop with (including himself), Nathan has to pay an extra <math>\$1</math> for gas to drive everyone to the shop. The gift that Nathan buys costs <math>\$100</math>, and everybody shopping with him splits the cost of the gift evenly. How many people should Nathan bring with him (excluding himself) to minimize the total cost he needs to pay for the drive and the gift?
 +
 +
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 9|Solution]]
 +
 +
===Problem 10===
 +
 +
Define a sequence <math>a_1 = 0</math>, <math>a_2 = 1</math>, and <math>a_n = 3a_{n-1} - 2a_{n-2}</math> for <math>n \geq 3</math>. Find the largest integer <math>n</math>
 +
such that <math>2^n</math> divides <math>a_{100} + 1</math>.
 +
 +
<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 99\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 101</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 10|Solution]]
 +
 +
===Problem 11===
 +
 +
At a school, there exist four student council officers: the president, the vice president, the treasurer, and the secretary. When the officers sit in four chairs in a line, the treasurer and the president cannot sit next to each other, or they will begin to talk and no work will be done. Given this condition, what is the probability the vice president and the president sit next to each other?
 +
 +
<math>\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4}</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 11|Solution]]
 +
 +
===Problem 12===
 +
 +
Find the number of <math>x</math> such that <math>|||||x - 1| - 1| - 1| - 1| - 1| - \frac{1}{2} = 0</math>
 +
 +
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 32</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 12|Solution]]
 +
 +
===Problem 13===
 +
 +
Given that there is exactly one integer such that fourty-eight multiplied by the square of the reciprocal of the integer added onto the integer itself is equal to seven. Find the number of factors of this integer.
 +
 +
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 13|Solution]]
 +
 +
===Problem 14===
 +
 +
What is the sum of <math>a + b + c</math> for all triples of positive integers <math>a < b < c</math> such that <math>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1</math>?
 +
 +
<math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 14|Solution]]
 +
 +
===Problem 15===
 +
 +
Shreyan and Kaylee are on opposite vertices of a cube. Each turn, they both randomly move across an adjacent edge to another vertex with equal probability. What is the expected number of turns that occur until the two meet in the middle of an edge for the first time?
 +
 +
<math>\textbf{(A)}\ 13\qquad\textbf{(B)}\ 13.5\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 26</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 15|Solution]]
 +
 +
===Problem 16===
 +
 +
Let <math>(x,y)</math> be real numbers which satisfy <math>(x-48)^2+(y-14)^2=169</math>. What is the maximum possible value of <math>\sqrt{x^2+y^2}</math>?
 +
 +
<math>\textbf{(A)}\ 26\qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 16|Solution]]
 +
 +
===Problem 17===
 +
 +
Alice and Bob are playing a series of games where Alice never has three more wins than Bob and Bob never has three more wins than Alice. Games never end in a draw. The number of ways for them to play <math>20</math> games such that the series ends in a draw can be written as <math>a^xb^y</math>, where <math>a</math> and <math>b</math> are distinct primes and <math>x</math> and <math>y</math> are positive integers. What is <math>a + b + x + y</math>?
 +
 +
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 17|Solution]]
 +
 +
===Problem 18===
 +
 +
Let <math>AB = 6</math> be the diameter of a circle. Let <math>C</math> vary along the circle, and let <math>S</math> be the set of all
 +
possible incenters of <math>\triangle ABC</math>. <math>S</math> is a curve missing two points. When the two points are added to
 +
<math>S</math>, the area enclosed by <math>S</math> can be written as <math>a\pi - b</math>. Find <math>a + b</math>.
 +
 +
<math>\textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 27\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 18|Solution]]
 +
 +
===Problem 19===
 +
 +
The city of Dallas is issuing new <math>6</math>-digit license plates. To make keeping track of different license places easier, each license plate much differ from every other license plate by at least <math>2</math> digits. For example, the license plates <math>012345</math> and <math>012344</math> may not both be issued, while the license plates <math>012345</math> and <math>012354</math> may. What is the maximum number of license plates that can be issued?
 +
 +
<math>\textbf{(A)}\ 10,000\qquad\textbf{(B)}\ 20,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 19|Solution]]
 +
 +
===Problem 20===
 +
 +
If
 +
\begin{align*}
 +
a+8b+27c+64d=2,\\
 +
8a+27b+64c+125d=20,\\
 +
27a+64b+125c+216d=202,\\
 +
64a+125b+216c+343d=2024,\\
 +
\end{align*}
 +
then find the remainder when
 +
<cmath>125a+216b+343c+512d</cmath>
 +
is divided by <math>1000</math>.
 +
 +
<math>\textbf{(A)}\ 222\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 476\qquad\textbf{(D)}\ 938\qquad\textbf{(E)}\ 962</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 20|Solution]]
 +
 +
===Problem 21===
 +
 +
Given that <math>r,s,t</math> are the three solutions of <math>x^3-2x^2-3x+5=0</math>, find <math>r^4+s^4+t^4</math>.
 +
 +
<math>\textbf{(A)}\ 38\qquad\textbf{(B)}\ 39\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 42</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 21|Solution]]
 +
 +
===Problem 22===
 +
 +
In triangle <math>ABC</math> with <math>AC=3</math> and <math>BC=5</math> and a right angle at <math>A</math>, circle <math>c_1</math> is centered at <math>B</math> and passes through <math>A</math>. Circle <math>c_2</math> lies inside the triangle and is tangent to <math>AB</math>, <math>BC</math>, and <math>c_1</math>. Given that the radius of <math>c_2</math> can be expressed as <math>\frac{a\sqrt{b}-c}{d}</math> for positive integers <math>a,b,c,d</math> and square-free <math>b</math>, what is the sum <math>a+b+c+d</math>?
 +
 +
<math>\textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 27\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 31</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 22|Solution]]
 +
 +
===Problem 23===
 +
 +
How many positive integers less than or equal to <math>5000</math> can be written in the form <math>m^2+n^8</math>, where <math>m</math> and <math>n</math> are positive integers?
 +
 +
<math>\textbf{(A)}\ 134\qquad\textbf{(B)}\ 135\qquad\textbf{(C)}\ 136\qquad\textbf{(D)}\ 137\qquad\textbf{(E)}\ 138</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 23|Solution]]
 +
 +
===Problem 24===
 +
 +
There are <math>5</math> people standing in a line, with their heights being some permutation of the integers <math>1,2,/cdots,5</math>. Some people however, are unhappy because they can’t see the front of the line as there is someone blocking them. As a result, we define the extension needed of each person to be the minimum extra height they would need such that no one in front of them would block them. This is defined independently for each person. Find the expected sum of the extensions needed.
 +
 +
<math>\textbf{(A)}\ \frac{61}{10}\qquad\textbf{(B)}\ \frac{31}{5}\qquad\textbf{(C)}\ \frac{25}{4}\qquad\textbf{(D)}\ \frac{63}{10}\qquad\textbf{(E)}\ \frac{34}{5}</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 24|Solution]]
 +
 +
===Problem 25===
 +
 +
Triangle <math>ABC</math> with <math>AB=14</math>, <math>BC=13</math>, and <math>AC=15</math> is inscribed in a circle. Let <math>D</math> be a point on minor arc <math>BC</math> such that quadrilateral <math>ABCD</math> has an inscribed circle. If the length of <math>BD</math> can be expressed as <math>\frac{a+\sqrt{b}}{c}</math> where <math>a,b,c</math> are integers, <math>\gcd(a,b,c)=1</math>, and <math>b</math> is squarefree, find <math>b</math>.
 +
 +
<math>\textbf{(A)}\ 211\qquad\textbf{(B)}\ 274\qquad\textbf{(C)}\ 362\qquad\textbf{(D)}\ 401\qquad\textbf{(E)}\ 434</math>
 +
 +
[[2024 DMC Mock 10 Problems/Problem 25|Solution]]

Revision as of 20:14, 16 September 2024

Problem 1

Compute the value of $8\left(\frac{2}{13}+\frac{2}{15}\right)+2\left(\frac{5}{13}+\frac{7}{15}\right)$

$\textbf{(A)}\ \frac{40}{13}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{72}{65}\qquad\textbf{(E)}\ \frac{52}{15}$

Solution

Problem 2

Since Branden Kim is the paragon of all human emotion, he is most resplendent in love and accolades. Who is Branden Kim? For the blind, he is their vision. For the starving, he is their nourishment. For the thirsty, he is their water. For the depressed, he is their happiness. For the oppressed, he is their salvation. He will stand up to fight all injustice. Even though he is only one hundred fifty centimeters tall, he is the champion who blocks all injustice. If Branden Kim has one million fans, I am one of them. If Branden Kim has a hundred fans, I am one of them. If Branden Kim only has one fan, then that is me. If Branden Kim has no fans, I no longer exist. If the world is for Branden Kim, I am for the world. If the world is against Branden Kim, I am against the world. That being said, please, with all due respect, tell me how close the great Branden Kim is to the heavens in meters, assuming the heavens are $1000$ meters off the ground.

$\textbf{(A)}\ 850\qquad\textbf{(B)}\ 9885\qquad\textbf{(C)}\ 9985\qquad\textbf{(D)}\ 985\qquad\textbf{(E)}\ 998.5$

Solution

Problem 3

It takes $15$ minutes for Alice to deliver a cake. If Alice needs to deliver $10$ cakes and she starts delivering cakes at $1:00$, what time will she finish?

$\textbf{(A)}\ 1:30\qquad\textbf{(B)}\ 2:30\qquad\textbf{(C)}\ 2:50\qquad\textbf{(D)}\ 3:15\qquad\textbf{(E)}\ 3:30$

Solution

Problem 4

When $32, 47,$ and $77$ are divided by a positive integer $n$, the remainder is the same for all three divisions. What is the greatest possible value of $n$?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 30$

Solution

Problem 5

Ten logicians are sitting at a table. A server comes and asks if everyone wants coffee. The first logician answers “I don’t know.” Then the second logician answers “I don’t know.” This continues, with each logician answering “I don’t know,” until the tenth logician answers “no, not everyone wants coffee.” How many of the ten logicians want coffee?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Solution

Problem 6

Alice, Bob, and Charlie are sharing $15$ identical candies. Because Bob is greedy, he insists that he gets at least $5$ candies. Find the number of ways to distribute the candies.

$\textbf{(A)}\ 55\qquad\textbf{(B)}\ 66\qquad\textbf{(C)}\ 78\qquad\textbf{(D)}\ 91\qquad\textbf{(E)}\ 105$

Solution

Problem 7

Bob is advertising the Dallas Reunion Tower by making a poster comparing its height to the Burj Khalifa. Currently, in his diagram, the image of the Burj Khalifa is five times as tall as the Reunion Tower. Bob wants to scale the image of the Reunion tower so that it is $90\%$ the height of the Burj Khalifa. If the area of the image of the Reunion tower was originally $100$ square inches, what is the area, in square inches, of the scaled image? (Note that scaling is done proportionately in both width and length).

$\textbf{(A)}\ 450\qquad\textbf{(B)}\ 1000\qquad\textbf{(C)}\ 2025\qquad\textbf{(D)}\ 5000\qquad\textbf{(E)}\ 8100$

Solution

Problem 8

Compute $2020^2 + 2024^2 - 2020 \cdot 2024 + 2027^2 - 2027(2020 + 2024)$.

$\textbf{(A)}\ 36\qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 38\qquad\textbf{(D)}\ 39\qquad\textbf{(E)}\ 40$

Solution

Problem 9

Nathan and his friends are shopping together for a birthday gift for Nathan’s new girlfriend to celebrate Nathan’s success. For each person he brings along to shop with (including himself), Nathan has to pay an extra $$1$ for gas to drive everyone to the shop. The gift that Nathan buys costs $$100$, and everybody shopping with him splits the cost of the gift evenly. How many people should Nathan bring with him (excluding himself) to minimize the total cost he needs to pay for the drive and the gift?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Solution

Problem 10

Define a sequence $a_1 = 0$, $a_2 = 1$, and $a_n = 3a_{n-1} - 2a_{n-2}$ for $n \geq 3$. Find the largest integer $n$ such that $2^n$ divides $a_{100} + 1$.

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 99\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 101$

Solution

Problem 11

At a school, there exist four student council officers: the president, the vice president, the treasurer, and the secretary. When the officers sit in four chairs in a line, the treasurer and the president cannot sit next to each other, or they will begin to talk and no work will be done. Given this condition, what is the probability the vice president and the president sit next to each other?

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

Solution

Problem 12

Find the number of $x$ such that $|||||x - 1| - 1| - 1| - 1| - 1| - \frac{1}{2} = 0$

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 32$

Solution

Problem 13

Given that there is exactly one integer such that fourty-eight multiplied by the square of the reciprocal of the integer added onto the integer itself is equal to seven. Find the number of factors of this integer.

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$

Solution

Problem 14

What is the sum of $a + b + c$ for all triples of positive integers $a < b < c$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$?

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

Solution

Problem 15

Shreyan and Kaylee are on opposite vertices of a cube. Each turn, they both randomly move across an adjacent edge to another vertex with equal probability. What is the expected number of turns that occur until the two meet in the middle of an edge for the first time?

$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 13.5\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 26$

Solution

Problem 16

Let $(x,y)$ be real numbers which satisfy $(x-48)^2+(y-14)^2=169$. What is the maximum possible value of $\sqrt{x^2+y^2}$?

$\textbf{(A)}\ 26\qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65$

Solution

Problem 17

Alice and Bob are playing a series of games where Alice never has three more wins than Bob and Bob never has three more wins than Alice. Games never end in a draw. The number of ways for them to play $20$ games such that the series ends in a draw can be written as $a^xb^y$, where $a$ and $b$ are distinct primes and $x$ and $y$ are positive integers. What is $a + b + x + y$?

$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19$

Solution

Problem 18

Let $AB = 6$ be the diameter of a circle. Let $C$ vary along the circle, and let $S$ be the set of all possible incenters of $\triangle ABC$. $S$ is a curve missing two points. When the two points are added to $S$, the area enclosed by $S$ can be written as $a\pi - b$. Find $a + b$.

$\textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 27\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36$

Solution

Problem 19

The city of Dallas is issuing new $6$-digit license plates. To make keeping track of different license places easier, each license plate much differ from every other license plate by at least $2$ digits. For example, the license plates $012345$ and $012344$ may not both be issued, while the license plates $012345$ and $012354$ may. What is the maximum number of license plates that can be issued?

$\textbf{(A)}\ 10,000\qquad\textbf{(B)}\ 20,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000$

Solution

Problem 20

If \begin{align*} a+8b+27c+64d=2,\\ 8a+27b+64c+125d=20,\\ 27a+64b+125c+216d=202,\\ 64a+125b+216c+343d=2024,\\ \end{align*} then find the remainder when \[125a+216b+343c+512d\] is divided by $1000$.

$\textbf{(A)}\ 222\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 476\qquad\textbf{(D)}\ 938\qquad\textbf{(E)}\ 962$

Solution

Problem 21

Given that $r,s,t$ are the three solutions of $x^3-2x^2-3x+5=0$, find $r^4+s^4+t^4$.

$\textbf{(A)}\ 38\qquad\textbf{(B)}\ 39\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 42$

Solution

Problem 22

In triangle $ABC$ with $AC=3$ and $BC=5$ and a right angle at $A$, circle $c_1$ is centered at $B$ and passes through $A$. Circle $c_2$ lies inside the triangle and is tangent to $AB$, $BC$, and $c_1$. Given that the radius of $c_2$ can be expressed as $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ and square-free $b$, what is the sum $a+b+c+d$?

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 27\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 31$

Solution

Problem 23

How many positive integers less than or equal to $5000$ can be written in the form $m^2+n^8$, where $m$ and $n$ are positive integers?

$\textbf{(A)}\ 134\qquad\textbf{(B)}\ 135\qquad\textbf{(C)}\ 136\qquad\textbf{(D)}\ 137\qquad\textbf{(E)}\ 138$

Solution

Problem 24

There are $5$ people standing in a line, with their heights being some permutation of the integers $1,2,/cdots,5$. Some people however, are unhappy because they can’t see the front of the line as there is someone blocking them. As a result, we define the extension needed of each person to be the minimum extra height they would need such that no one in front of them would block them. This is defined independently for each person. Find the expected sum of the extensions needed.

$\textbf{(A)}\ \frac{61}{10}\qquad\textbf{(B)}\ \frac{31}{5}\qquad\textbf{(C)}\ \frac{25}{4}\qquad\textbf{(D)}\ \frac{63}{10}\qquad\textbf{(E)}\ \frac{34}{5}$

Solution

Problem 25

Triangle $ABC$ with $AB=14$, $BC=13$, and $AC=15$ is inscribed in a circle. Let $D$ be a point on minor arc $BC$ such that quadrilateral $ABCD$ has an inscribed circle. If the length of $BD$ can be expressed as $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers, $\gcd(a,b,c)=1$, and $b$ is squarefree, find $b$.

$\textbf{(A)}\ 211\qquad\textbf{(B)}\ 274\qquad\textbf{(C)}\ 362\qquad\textbf{(D)}\ 401\qquad\textbf{(E)}\ 434$

Solution

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