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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]]
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===Problem 4===
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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>?
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<math>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 30</math>
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[[2024 DMC Mock 10 Problems/Problem 4|Solution]]
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===Problem 5===
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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?
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<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math>
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[[2024 DMC Mock 10 Problems/Problem 5|Solution]]
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===Problem 6===
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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.
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<math>\textbf{(A)}\ 55\qquad\textbf{(B)}\ 66\qquad\textbf{(C)}\ 78\qquad\textbf{(D)}\ 91\qquad\textbf{(E)}\ 105</math>
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[[2024 DMC Mock 10 Problems/Problem 6|Solution]]
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===Problem 7===
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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).
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<math>\textbf{(A)}\ 450\qquad\textbf{(B)}\ 1000\qquad\textbf{(C)}\ 2025\qquad\textbf{(D)}\ 5000\qquad\textbf{(E)}\ 8100</math>
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[[2024 DMC Mock 10 Problems/Problem 7|Solution]]

Revision as of 18:50, 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

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