# Nov 2018 (Mathcounts)

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.

Week of Nov 25 – Dec 1: Expected Values (from the 2019 Mathcounts School Handboook)

Nov 25: A fair 10-sided die with one face labeled 1, two faces labeled 2, three faces labeled 3 and four faces labeled 4 is rolled. What is the expected value when this die is rolled?

Answer: 1*(1/10) + 2*(2/10) + 3*(3/10) + 4*(4/10) = 30/10 = 3

Nov 26: Ana has a bowl containing two square tiles, one with side length 2 cm and the other with side length 3 cm. She randomly chooses a tile from the bowl. The expected value of the area of the chosen tile is the sum of the products of each tile’s area and its corresponding probability of being chosen. If the probability of choosing a particular tile is proportional to its area, what is the expected value of the area of the tile Ana chooses? Express your answer as a common fraction.

Answer: The probability of choosing the 2×2 tile is 4/13 and the probability of choosing the 3×3 tile is 9/13. Expected area = 4*(4/13) + 9*(9/13) = 97/13.

Nov 27: Gwen randomly draws a card from a deck of 40 cards numbered 1 through 40. What is the expected value of the number on the card she draws? Express your answer as a decimal to the nearest tenth.

Nov 28: Luke paints each face of a 5 × 5 × 5 cube red. He then cuts the cube into 125 unit cubes and randomly chooses a single unit cube. What is the expected value of the number of painted faces on this unit cube? Express your answer as a decimal to the nearest tenth.

Answer: 8 cubes with 3 painted faces, 36 cubes with 2 painted faces, 54 cubes with 1 painted faces. 3*(8/125) + 2*(36/125) + 1*(54/125) = 150/125 = 1.2

Nov 29: In each round of a particular game, Dinara can win at most one point. If she has a 70% chance of winning a point in each round, what is the expected value of Dinara’s total score after three rounds? Express your answer as a decimal to the nearest tenth.

Nov 30: Jo and her four friends each secretly pick a random integer from −5 to 5, inclusive. What is the expected value of the sum of the five chosen numbers?

Answer: 0. (The expected value that each person picks is 0.)

Dec 1: Allen randomly distributes 1000 jelly beans into 10 jars lined up in a row from left to right. What is the expected value of the number of jelly beans in the leftmost jar?

Answer: 100 (Each bean has 1/10 of probability to be in the leftmost jar.)

Week of Nov 18 – Nov 24: Measurements (from the 2019 Mathcounts School Handboook)

Nov 18: Merri places weights of 6 units and 28 units on the right side of a balance and weights of 3 units and 19 units on the left side. If she adds an object to the left side that makes the balance level, how many units does the object weigh?

Nov 19: The weight of a small clip is 2/3 the weight of a large clip. If 2 tacks weigh the same amount as a large clip, how many tacks weigh the same amount as 12 small clips?

Nov 20: On the planet Klem, 1 Bem plus 7 Dems equals 4 Pems, and 2 Bems plus 1 Dem equals 1 Pem. How many Dems equal 7 Bems?

Nov 21: If a race car is traveling at 99 mi/h, how many meters does it travel in a second, given that 0.305 meter = 1 foot? Express your answer as a decimal to the nearest tenth.

Answer: 99 * 5280 * 0.305 / 3600 =  44.3 m/s.

Nov 22: If the results when reading a measuring stick can be off by at most 1 cm, what is
the maximum percent error when 24 cm is measured? Express your answer to the
nearest tenth.

Nov 23: Vijay gives Sanjay a set of four weights of 1, 3, 8 and 26 grams. When Sanjay
places weights on either side of a balance, what is the smallest positive integer number of grams that he cannot measure with this set?

Nov 24: If Clem has 2 cups, 7 pints, 8 quarts and 11 half-gallons of lemonade, how many total gallons of lemonade does she have? Express your answer as a mixed number.

Answer:  1 gallon = 4 quarts = 8 pints = 16 cups.   2/16 + 7/8 + 8/4 + 11/2 = (1+7+16+44)/8 = 8 1/2

Here’s a friendly reminder that the AMC8 will be on Tue Nov 13. Please contact Professor Louis Beaugris <lbeaugri@kean.edu> who has hosted the test for many years at Kean University. The test has 25 multiple choice questions to be completed in 40 minutes. Here’s the link to last year’s exam if you would like to check it out:
https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems

Week of Nov 11 – Nov 17: Selected problems from AMC8 2014. Since we haven’t practiced geometry so much this year, I’m including more geometry problems here.

Nov 11: Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book? $\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$

Nov 12: Eleven members of the Middle School Math Club each paid the same integer amount for a guest speaker to talk about problem solving at their math club meeting. In all, they paid their guest speaker $\textdollar\underline{1}\underline{A}\underline{2}$. What is the missing digit A of this 3-digit number? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Answer: D. The number is a multiple of 11.

Nov 13: In $\bigtriangleup ABC$ $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$? $[asy] size(300); defaultpen(linewidth(0.8)); pair A=(-1,0),C=(1,0),B=dir(40),D=origin; draw(A--B--C--A); draw(D--B); dot("A", A, SW); dot("B", B, NE); dot("C", C, SE); dot("D", D, S); label("70^\circ",C,2*dir(180-35));[/asy]$ $\textbf{(A) }100\qquad\textbf{(B) }120\qquad\textbf{(C) }135\qquad\textbf{(D) }140\qquad \textbf{(E) }150$

Nov 14: Rectangle ABCD and right triangle DCE have the same area. They are joined to form a trapezoid, as shown. What is DE? $[asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label("A",A,NW); label("B",B,SW); label("C",C,S); label("D",D,N); label("E",E,S); label("5",A/2,W); label("6",(A+D)/2,N); [/asy]$ $\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }16$

Answer: B. CE = 12, CD = 5, DE = 13.

Nov 15: Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely? $\textbf{(A) }$ all 4 are boys $\textbf{(B) }$ all 4 are girls $\textbf{(C) }$ 2 are girls and 2 are boys $\textbf{(D) }$ 3 are of one gender and 1 is of the other gender $\textbf{(E) }$ all of these outcomes are equally likely

Answer: D. Prob[4 boys] = Prob[4 girls] = 1/32. Prob[2 girls and 2 boys] = 6/32. Prob[3 boys and 1girl, or 3 girls and 1 boy] = 2*4/32

Nov 16: Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles? $[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E);[/asy]$ $\textbf{(A) }3.5\qquad\textbf{(B) }4.0\qquad\textbf{(C) }4.5\qquad\textbf{(D) }5.0\qquad\textbf{(E) }5.5$

Nov 17: Three members of the Euclid Middle School girls’ softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all 2-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That’s funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today’s date.

What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad\textbf{(E) }23$

Answer: A.  20<A+C<B+C<A+B<32. Therefore, C<A<B. The only prime C can have is 11.

Week of Nov 4 – Nov 10: Selected problems from AMC8 2016.

Nov 4: Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score? $\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$

Nov 5: Which of the following numbers is not a perfect square? $\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}$

Answer: B (If the exponent is even, then the number is a perfect square. D can be rewritten with base 2, in which case the exponent is even.)

Nov 6: Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $$2 * (5 * x)=1$$ $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

Nov 7: Karl’s car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day? $\textbf{(A)}\mbox{ }525\qquad\textbf{(B)}\mbox{ }560\qquad\textbf{(C)}\mbox{ }595\qquad\textbf{(D)}\mbox{ }665\qquad\textbf{(E)}\mbox{ }735$

Nov 8: An ATM password at Fred’s Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible? $\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\textbf{(E)}\mbox{ }9999$

Nov 9: The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$

Nov 10: A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle? $[asy]draw((0,0)--(8,15)--(16,0)--(0,0)); draw(arc((8,0),7.0588,0,180));[/asy]$ $\textbf{(A) }4 \sqrt{3}\qquad\textbf{(B) } \dfrac{120}{17}\qquad\textbf{(C) }10\qquad\textbf{(D) }\dfrac{17\sqrt{2}}{2}\qquad \textbf{(E)} \dfrac{17\sqrt{3}}{2}$