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Difference between revisions of "DMC Mock AMC 10"
Line 57: Line 57:
 
===Problem 8===
 
===Problem 8===
  
−
Compute <math>2020^2 + 2024^2 − 2020 \cdot 2024 + 2027^2 − 2027(2020 + 2024)</math>.
+
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>
 
<math>\textbf{(A)}\ 36\qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 38\qquad\textbf{(D)}\ 39\qquad\textbf{(E)}\ 40</math>
Line 73: Line 73:
 
===Problem 10===
 
===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>
+
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>.
 
such that <math>2^n</math> divides <math>a_{100} + 1</math>.
  

Revision as of 19:55, 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

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