# Feb 2017

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.

Since a few new mathletes are younger than the rest, I decided to post 2 versions of the practice problems.  Version $\alpha$ contains new problems, and version $\beta$ pulls out past problems from the Archive. The purpose of version $\beta$ is to offer new mathletes an option of easier problems.

Answers: B, D, B, C, D, B ,A.

Version $\alpha$: This is our last week.  Some of you just competed in MATHCOUNTS and will take the AMC10 test next week.  We’ll finish off with 7 probability problems from AMC10.

Feb 5: Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length?

$\textbf{(A) }\frac{2}5\qquad\textbf{(B) }\frac{4}9\qquad\textbf{(C) }\frac{1}2\qquad\textbf{(D) }\frac{5}9\qquad\textbf{(E) }\frac{4}5$

Feb 6: A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$

Feb 7: Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

$\textbf {(A) } \frac{1}{36} \qquad \textbf {(B) } \frac{7}{72} \qquad \textbf {(C) } \frac{1}{9}\qquad \textbf {(D) } \frac{5}{36} \qquad \textbf {(E) } \frac{1}{6}$

Feb 8: Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al’s number will be a whole number multiple of Bill’s and Bill’s number will be a whole number multiple of Cal’s?

$\textbf{(A) } \dfrac{9}{1000} \qquad\textbf{(B) } \dfrac{1}{90} \qquad\textbf{(C) } \dfrac{1}{80} \qquad\textbf{(D) } \dfrac{1}{72} \qquad\textbf{(E) } \dfrac{2}{121}$

Feb 9: Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even?

$\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

Feb 10: Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?

$\textbf{(A) } \dfrac{8}{99} \qquad\textbf{(B) } \dfrac{2}{5} \qquad\textbf{(C) } \dfrac{9}{20} \qquad\textbf{(D) } \dfrac{1}{2} \qquad\textbf{(E) } \dfrac{9}{16}$

Feb 11: Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is $0$.)

$\mathrm{(A)}\ {{{\frac{3} {8}}}} \qquad \mathrm{(B)}\ {{{\frac{1} {2}}}} \qquad \mathrm{(C)}\ {{{\frac{43} {72}}}} \qquad \mathrm{(D)}\ {{{\frac{5} {8}}}} \qquad \mathrm{(E)}\ {{{\frac{2} {3}}}}$

Version $\beta$:  This week we will focus on problems related to volume. Again, you are allowed to use calculator if “Calculator” is indicated.

Answers: 4, 3, 1207, $144w^3$, 4576, $72\sqrt{2}$, 1/27

Feb 5. Two cylinders are equal in volume. The radius of one is doubled, and the height of the other cylinder is increased to k times its original height. If the two new cylinders are equal in volume, what is the value of k?

Feb 6. The surface area of a sphere, in square meters, and its volume, in cubic meters, are numerically equal. What is the length of the radius of the sphere?

Feb 7. Boynton’s sheet cake measures 18 × 24 inches and has a height of 4 inches. However, these measurements include a 3/4-inch thick layer of frosting on the top and sides. What is the volume of Boynton’s cake excluding the frosting? Express your answer to the nearest whole number. (Calculator)

Feb 8. A certain box of width w has a length that is twice its width, and its height is three times its width. What is the total volume of 24 of these boxes? Express your answer in terms of w.

Feb 9. For a particular rectangular solid with integer dimensions, the sum of its length, width and height is 50 cm. What is the absolute difference between the greatest possible volume and the least possible volume of the solid?

Feb 10. A pyramid with a square base of side length 6 m has a height equal to the length of the diagonal of the base. What is the volume of the pyramid? Express your answer in simplest radical form.

Feb 11. Connecting the centers of the four faces of a regular tetrahedron creates a smaller regular tetrahedron. What is the ratio of the volume of the smaller tetrahedron to the volume of the original one? Express your answer as a common fraction.

Version $\alpha$:   This week we will explore the technique called “stars and sticks”.  Last week we had the following problem:

How many solutions to $x+y+z=n$ are there in terms of $n$, if $x, y$ and $z$ are nonnegative integers?

We use the “stars and sticks” technique to solve this problem.  We line up $n$ stars and we will put down 2 sticks to separate these stars into 3 groups.  The number of stars before the stick 1 is the value of x; the number of stars between sticks 1 and 2 is the value of y; and the number of stars after stick 2 is the value of z.  Note that two sticks can be next to each other with no stars in between, which allows y to be 0.  There can also be no stars before stick 1 which allows x to be 0, or no stars after stick 2 which allows z to be zero.

So how many ways we can place 2 sticks to separate $n$ stars?  It is ${{n+2}\choose 2}$ because there are n+2 places altogether, n for stars and 2 for sticks.

If we have more variables than x, y and z, the total number of nonnegative solutions is ${{n+m-1}\choose {m-1}}$ for m variables. This is because $m-1$ sticks can create m groups, one for each variable.

Let us practice the stars and sticks technique.

Jan 29: I have 12 identical pieces of candies. In how many ways can I give them to 4 kids? (Hint: 12 candies are 12 stars, and 3 sticks will be put down to partition the 12 stars into 4 groups.)
[Answer] ${{12+4-1}\choose{4-1}}= {15\choose 3}$

Jan 30: I have 12 identical pieces of candies. In how many ways can I give them to 4 kids so that every child has at least 1 piece? (Hint: We give 1 kid one piece to begin with, and then distribute 8 candies to 4 kids.)
[Answer] ${{8+4-1}\choose{4-1}}={11\choose 3}$

Jan 31: I have 12 identical pieces of candies. In how many ways can I give them to 4 kids so that every child has at least 2 pieces?
[Answer] ${{4+4-1}\choose{4-1}}={7\choose 3}$

Feb 1: I have 12 identical pieces of candies. In how many ways can I give them to 4 kids so that the oldest child gets at least 2 candies?
[Answer] ${{10+4-1}\choose{4-1}}={13\choose 3}$

Feb 2: I have 5 pieces of chocolate candies and 7 pieces of butterscotch candies. In how many ways can I give them to 5 kids?
[Answer] ${{5+5-1}\choose{5-1}}{{7+5-1}\choose{5-1}}= {9\choose 4}{11\choose 4}$

Feb 3: How many integer solutions does $x_1+x_2+x_3+x_4=20$ have if $x_1\ge 1$, $x_2\ge 2$, $x_3\ge 3$ and $x_4\ge 4$?
[Answer] ${{10+4-1}\choose{4-1}}={13\choose 3}$

Feb 4How many positive integer solutions does $3x_1+x_2+x_3+x_4=20$ have?
[Answer] This is done through case study.  If $x_1=1$, $x_2+x_3+x_4=17$ which has ${{14+3-1}\choose{3-1}}={16\choose 2}$ solutions.  If $x_1=2$, $x_2+x_3+x_4=14$, which has ${{11+3-1}\choose{3-1}}$ solutions.  If $x_1=3$, …  If $x_1=5$… you get the idea.

Version $\beta$:   This week we will focus on problems related to area. You are allowed to use calculator if “Calculator” is indicated.

Answers: 79, 27.7, 60, 400, 682, 30, 16

Jan 29. What is the area of a circle that circumscribes a 6-cm by 8-cm rectangle? Express your answer to the nearest whole number.   (Calculator)

Jan 30. A pizza of diameter 16 inches has an area large enough to serve 4 people. What is the diameter of a pizza with an area large enough to serve 12 people? Express your answer as a decimal to the nearest tenth. (Calculator)

Jan 31. What is the area of triangle ABC with vertices A(2, 3), B(17, 11) and C(17, 3) ?

Feb 1. A square residential lot is measured to be 100 feet on each side, with a possible measurement error of 1% in each of the length and width. What is the absolute difference between the largest and smallest possible measures of the area given this possible error? (Calculator)

Feb 2. Five rectangles are arranged in a row. Each rectangle is half as tall as the previous one. Also, each rectangle’s width is half its height. The first rectangle is 32 cm tall. What is the sum of the areas of all five rectangles?

Feb 3. The amount of paint required to cover a surface is directly proportional to the area of that surface. The amount of paint needed to cover five spheres of radius 10 inches is the same as the amount of paint required to cover a solid right circular cylinder with radius 20 inches. What is the height of the cylinder?

Feb 4. ABCD is a square with side length 4 units, and AEFC is a rectangle with point B on side EF. What is the area of AEFC?