Nov 2017 (AMC 8)

I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday.  For those of you who practiced with me last year, the process is similar.

For instructions on submitting weekly answers, click on the “Logistics” tab.
For previously posted problems, click on the “Archive” tab.
For previous prizes, click on the “News” tab.

AMC 8 is next Tue.  I picked 7 problems from the 2007 test.  Try to do them in 1 sitting and see how long it takes you to finish.

Click GoogleForm to submit answers by Sat, Nov 18.

Week of Nov 12 – Nov 18.

Nov 12: $650$ students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

AMC8 2007 2.png$\mathrm{(A)} \frac{2}{5} \qquad \mathrm{(B)} \frac{1}{2} \qquad \mathrm{(C)} \frac{5}{4} \qquad \mathrm{(D)} \frac{5}{3} \qquad \mathrm{(E)} \frac{5}{2}$

Answer: E.  250/100

Nov 13: The average cost of a long-distance call in the USA in $1985$ was $41$ cents per minute, and the average cost of a long-distance call in the USA in $2005$ was $7$ cents per minute. Find the approximate percent decrease in the cost per minute of a long- distance call.

$\mathrm{(A)}\ 7 \qquad\mathrm{(B)}\ 17 \qquad\mathrm{(C)}\ 34 \qquad\mathrm{(D)}\ 41 \qquad\mathrm{(E)}\ 80$

Answer: E.  (41-7)/41 

Nov 14: For any positive integer $n$, define $\boxed{n}$ to be the sum of the positive factors of $n$. For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\boxed{\boxed{11}}$ .

$\mathrm{(A)}\ 13 \qquad \mathrm{(B)}\ 20 \qquad \mathrm{(C)}\ 24 \qquad \mathrm{(D)}\ 28 \qquad \mathrm{(E)}\ 30$

Answer: D.  Box 11 = 1+11 = 12.  Box 12 = 1+2+3+4+6+12 = 28 

Nov 15: A unit hexagram is composed of a regular hexagon of side length $1$ and its $6$ equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

AMC8 2007 12.png$\mathrm{(A)}\ 1:1 \qquad \mathrm{(B)}\ 6:5  \qquad \mathrm{(C)}\ 3:2 \qquad \mathrm{(D)}\ 2:1 \qquad \mathrm{(E)}\ 3:1$

Answer: A

Nov 16: Let $a, b$ and $c$ be numbers with $0 < a < b < c$. Which of the following is impossible?

$\mathrm{(A)} \ a + c < b  \qquad \mathrm{(B)} \ a \cdot b < c \qquad \mathrm{(C)} \ a + b < c \qquad \mathrm{(D)} \ a \cdot c < b \qquad \mathrm{(E)}\frac{b}{c} = a$

Answer: A 

Nov 17:  The product of the two $99$-digit numbers

$303,\!030,\!303,\!...,\!030,\!303$ and $505,\!050,\!505,\!...,\!050,\!505$

has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?

$\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ 5 \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 8 \qquad \mathrm{(E)}\ 10$

Answer: D.  Units digit 5, thousands digit, 3.  303×505

Nov 18: Two cards are dealt from a deck of four red cards labeled $A, B, C, D$ and four green cards labeled $A, B, C, D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?

$\mathrm{(A)} \frac{2}{7} \qquad \mathrm{(B)} \frac{3}{8} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{4}{7} \qquad \mathrm{(E)} \frac{5}{8}$

Answer: D.  Number of ways to choose a winning pair: 4 matching letters, 6+6=12 matching colors.  All possibilities: 8*7/2 = 28.  16/28 = 4/7

AMC 8 is coming up soon.  One difficult topic that we haven’t touched this year is probability, and so we’ll practice a few this week.  There is also an easy topic, problems related to time and dates.  We’ll also practice a few this week.

Click GoogleForm to submit answers by Sat, Nov 11.

Week of Nov 5 – Nov 11.

Nov 5:  A three-digit integer contains one of each of the digits $1$$3$, and $5$. What is the probability that the integer is divisible by $5$?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$

Answer: B.  Digit 5 has 1/3 of chance to be the unit digit. 

Nov 6: Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?

$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$

Answer: C. Probability of both picking green = 1/2*1/4; probability of both picking red = 1/2*2/4  

Nov 7: A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?

$\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac5{12}\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac7{12}\qquad\textbf{(E) }\dfrac56$

Answer: D. 6-6, 6-5, …6-1, 5-5, 5-4, …5-1, … 1-1.  So a total of 21 possible pairs out of 36 pairs.

Nov 8: Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?

$\textbf{(A) } \frac14 \qquad\textbf{(B) } \frac13 \qquad\textbf{(C) } \frac12 \qquad\textbf{(D) } \frac23 \qquad\textbf{(E) } \frac34$

Answer: B. Let Angie sit down in any seat.  Among the 3 remaining seats, 1 is opposite Angie.

Nov 9:  If February is a month that contains Friday the $13^{\text{th}}$, what day of the week is February 1?

$\textbf{(A)}\ \text{Sunday} \qquad \textbf{(B)}\ \text{Monday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday}\qquad \textbf{(E)}\ \text{Saturday}$

Answer: A. 

Nov 10: On February 13 $\emph{The Oshkosh Northwester}$ listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?


Answer: B

Nov 11: What time was it $2011$ minutes after midnight on January 1, 2011?

$\textbf{(A) }\text{January 1 at 9:31 PM}$

$\textbf{(B) }\text{January 1 at 11:51 PM}$

$\textbf{(C) }\text{January 2 at 3:11 AM}$

$\textbf{(D) }\text{January 2 at 9:31 AM}$

$\textbf{(E) }\text{January 2 at 6:01 PM}$

Answer: D.  Number of minutes per day = 24*60 = 1440. 2011-1440=571, which is 10 hours short of 29 minutes.