# Dec 2016

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.

Version $\alpha$,  Week of Dec. 25 – Dec. 31:  I noticed that Mathcounts often has problems that count the number of ways to form a committee. Some of such problems are easy while others can be messy. So let’s practice this type of counting problems.

Answers: ${20\choose 4} = 20*19*18*17/4! = 4845$; 20*19*18*17 = 116280; 20*19*18*17/2 = 58140;  ${5\choose 2} = 10$;  5*5 =25; ${5\choose 2} = 10$; ${5 \choose 2}+{5\choose 1}+{5\choose 0} = 10+5+1=16$

Dec 25: My school swimming club has 20 swimmers. How many ways can I swimmers to form a 4-person committee?

Dec 26: My school swimming club has 20 swimmers. How many ways can I swimmers to form a 4-person committee so that one is the captain, one is the vice-captain, one is the secretary and one is event coordinator?

Dec 27: My school swimming club has 20 swimmers. How many ways can I swimmers to form a 4-person committee so that one is the captain, one is the vice-captain and  two are secretaries?  The two secretary positions are not distinguishable.

Dec 28: A lock has 5 buttons numbered 1-5. It is opened by pushing 2 buttons simultaneously. How many possibilities are there?

Dec 29: A lock has 5 buttons numbered 1-5. It is opened by pushing 2 buttons one after another. How many possibilities are there?

Dec 30: A pizza shop offers 5 toppings, mushroom, pepper, meatball, sausage and pepperoni.  How many ways can 2 toppings be chosen?

Dec 31: A pizza shop offers 5 toppings, mushroom, pepper, meatball, sausage and pepperoni.  How many ways can at most 2 toppings be chosen?  (Extra credit: How many ways to choose at most 5 toppings? What if there are 10 toppings, and how many ways to choose at most 10 toppings?)

Version $\beta$,  Week of Dec. 25 – Dec. 31:   This week’s topic is geometry.

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

Dec 25:  Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is

$[asy] pair A,B,C,D; A = (5,5); B = (5,0); C = (0,0); D = (0,5); fill((0,0)--(0,4)--(1,4)--(1,1)--(4,1)--(4,0)--cycle,gray); draw(A--B--C--D--cycle); draw((4,0)--(4,4)--(0,4)); draw((1,5)--(1,1)--(5,1)); label($

$\text{(A)}\ 7 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Dec 26:  What is the degree measure of the smaller angle formed by the hands of a clock at 10 o’clock?

$[asy] draw(circle((0,0),2)); dot((0,0)); for(int i = 0; i < 12; ++i) { dot(2*dir(30*i)); } label($

$\text{(A)}\ 30 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90$

Dec 27: In triangle $CAT$, we have $\angle ACT = \angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ bisects $\angle ATC$, then $\angle CRT =$

$[asy] pair A,C,T,R; C = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570); draw(C--A--T--cycle); draw(T--R); label($

$\text{(A)}\ 36^\circ \qquad \text{(B)}\ 54^\circ \qquad \text{(C)}\ 72^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 108^\circ$

Dec 28: In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is

$[asy] draw((0,0)--(4,3)--(12,3)--(16,0)--cycle); draw((4,3)--(4,0),dashed); draw((3.2,0)--(3.2,.8)--(4,.8)); label($

$\text{(A)}\ 27 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 34 \qquad \text{(E)}\ 48$

Dec 29: Triangles $ABC$$ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?

$[asy] pair A,B,C,D,EE,F,G; A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3)); EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2); draw(A--B--C--cycle); draw(D--EE--A); draw(EE--F--G); label($

$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$

Source: AMC 2000

Dec 30: Given the areas of the three squares in the figure, what is the area of the interior triangle?

$[asy] draw((0,0)--(-5,12)--(7,17)--(12,5)--(17,5)--(17,0)--(12,0)--(12,-12)--(0,-12)--(0,0)--(12,5)--(12,0)--cycle,linewidth(1)); label($

$\mathrm{(A)}\ 13 \qquad\mathrm{(B)}\ 30 \qquad\mathrm{(C)}\ 60 \qquad\mathrm{(D)}\ 300 \qquad\mathrm{(E)}\ 1800$

Dec 31: The following figures are composed of squares and circles. Which figure has a shaded region with largest area?$[asy]/* AMC8 2003 #22 Problem */ size(3inch, 2inch); unitsize(1cm); pen outline = black+linewidth(1); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline); filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline); filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1)); filldraw(Circle((1,1), 1), white, outline); filldraw(Circle((3.5,.5), .5), white, outline); filldraw(Circle((4.5,.5), .5), white, outline); filldraw(Circle((3.5,1.5), .5), white, outline); filldraw(Circle((4.5,1.5), .5), white, outline); filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline); label($

$\textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}$

Click GoogleForm to submit answers. If you are doing version $\alpha$, you only have three solutions to submit.

Version $\alpha$,  Week of Dec. 18 – Dec. 24:  Let us finish off geometric probability. This week only has 3 problems, but I’d like you to submit your written solutions, not just the final numbers.  Two problems are from last week (Dec 14, Dec 17), since most submitted answers were incorrect.  One problem is the harder version of Dec 12 for which an answer cannot be guessed easily.

Dec 18 -19 [Please submit your written solution, not just the final answer.] A real number $x$ is chosen at random such that $0\le x\le 100$. What is the probability that $\lfloor \sqrt{x}\rfloor$ is even?

[Solution] For $64 \le x <81$, $\lfloor \sqrt{x} \rfloor = 8$.  For $36 \le x <49$, $\lfloor \sqrt{x} \rfloor = 6$. For $16 \le x < 25$, $\lfloor \sqrt{x} \rfloor = 4$. For $4 \le x <9$, $\lfloor \sqrt{x} \rfloor = 2$. For $0 \le x <1$, $\lfloor \sqrt{x} \rfloor = 0$.  So the length of the intervals is 17+13+9+5+1=45, and the probability is 45/100 = 9/20.

Dec 20-21 [Please submit your written solution, not just the final answer.] A dog is chained to a pole on a vertex of a regular hexagonal pen. The length of the chain equals the side length of the hexagon. If a chewing toy is thrown into the pen at random, what is the probability that the dog can reach the toy?

[Solution] Let the side length of the hexagonal pen be 1.  The area of the pen is then $6*(1/2)*1*\sqrt{3}/2=3\sqrt{3}/2$. The area in which the dog can reach the toy is 1/3 of a circle with radius 1, and this area is $\pi/3$.  So the probability is $2\pi/9\sqrt{3}$.

Dec 22-23 [Please submit your written solution, not just the final answer.]  A point P is chosen at random on a line segment $\bar{AB}$ of length 8. What is the probability that the distance from P to A is more than the square of the distance to B?  (We did a problem for line segment of length 6 last week.  For length 6 you could guess an answer, but not for length 8.)

[Solution] Let x be the distance from P to A.  Then $x> (8-x)^2$.  This simplifies to $x^2 -17x +64<0$, which solves to $(17-\sqrt{33})/2. So the probability is ${{8-(17-\sqrt{33})/2}\over 8} = {\sqrt{33}-1 \over 16}.$

Version $\beta$,  Week of Dec. 18 – Dec. 24: This week’s topic is about computing average.

Answers: D, D, C, E, A, D, B

Dec 18  Theresa’s parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks she helps around the house for $8$$11$$7$$12$ and $10$ hours. How many hours must she work for the final week to earn the tickets?

$\mathrm{(A)}\ 9 \qquad\mathrm{(B)}\ 10 \qquad\mathrm{(C)}\ 11 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 13$

Dec 19 The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people?

$\mathrm{(A)}\ 25 \qquad\mathrm{(B)}\ 26 \qquad\mathrm{(C)}\ 29 \qquad\mathrm{(D)}\ 33 \qquad\mathrm{(E)}\ 36$

Dec 20  The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:

        1. The largest isn’t first, but it is in one of the first three places.
2. The smallest isn’t last, but it is in one of the last three places.
3. The median isn’t first or last.


What is the average of the first and last numbers?

$\textbf{(A)}\ 3.5 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6.5 \qquad \textbf{(D)}\ 7.5 \qquad \textbf{(E)}\ 8$

Dec 21 Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?

$\text{(A)}\ \text{1 hr} \qquad \text{(B)}\ \text{1 hr 10 min} \qquad \text{(C)}\ \text{1 hr 20 min} \qquad \text{(D)}\ \text{1 hr 40 min} \qquad \text{(E)}\ \text{2 hr}$

Dec 22 Blake and Jenny each took four 100-point tests. Blake averaged 78 on the four tests. Jenny scored 10 points higher than Blake on the first test, 10 points lower than him on the second test, and 20 points higher on both the third and fourth tests. What is the difference between Jenny’s average and Blake’s average on these four tests?

$\mathrm{(A)}\ 10 \qquad\mathrm{(B)}\ 15 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 25 \qquad\mathrm{(E)}\ 40$

Dec 23 The average of the five numbers in a list is $54$. The average of the first two numbers is $48$. What is the average of the last three numbers?

$\textbf{(A)}\ 55\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 57\qquad\textbf{(D)}\ 58\qquad\textbf{(E)}\ 59$

Dec 24 There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is

$\text{(A)}\ 5\frac{3}{7} \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 6\frac{4}{7} \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 7\frac{3}{7}$

Click GoogleForm to submit answers by Dec 17.

Version $\alpha$,  Week of Dec 11 – Dec 17: Well, last week’s probability problems are hard.  So we step back and try a few easier ones this week.   Again, the basic formula is

prob[success] = number of successful events / number of total events
prob[success]  = size or successful region / size of total region

Answers: 1/2; 1/3; 1/6; $2\sqrt{3}\pi\over 27$;  $2\sqrt{3}/9$; 1/4; 9/20

Dec 11:  A point P is chosen at random on a line segment $\bar{AB}$ of length 6. What is the probability that P is closer to the midpoint of $\bar{AB}$ than to either end point?

Dec 12:  A point P is chosen at random on a line segment $\bar{AB}$ of length 6. What is the probability that the distance from P to A is more than the square of the distance to B?

Dec 13: A point P is chosen at random on the circumference of a regular hexagon ABCDEF. What is the probability that P is closer to vertex A than other vertices B, C, D, E and F?

Dec 14: A dog is chained to a pole on a vertex of a regular hexagonal pen. The length of the chain equals the side length of the hexagon. If a chewing toy is thrown into the pen at random, what is the probability that the dog can reach the toy?

Dec 15: A real number $x$ is chosen at random such that $-3 \le x \le 6$. What is the probability that $x^2 < 3$?

Dec 16: A real number $x$ is chosen at random such that $0\le x\le 100$. What is the probability that $x-\lfloor x\rfloor< 1/4$?  (Notation: $\lfloor x \rfloor$ denotes the integer when rounding down $x$.  For example $\lfloor 2.3\rfloor = 2$, $\lfloor 2 \rfloor= 2$, $\lfloor -2.3\rfloor =-3$.  So for a positive real number $x$,  $x-\lfloor x\rfloor$ is the decimal part of $x$.)

Dec 17: A real number $x$ is chosen at random such that $0\le x\le 100$. What is the probability that $\lfloor \sqrt{x}\rfloor$ is even?

Version $\beta$,  Week of Dec 11 – Dec 17:

Answers: B, A, D, B, C, D, E

Dec 11: For $x=7$, which of the following is the smallest?

$\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac{6}{x-1} \qquad \text{(D)}\ \dfrac{x}{6} \qquad \text{(E)}\ \dfrac{x+1}{6}$

Dec 12: $(6?3) + 4 - (2 - 1) = 5.$ To make this statement true, the question mark between the 6 and the 3 should be replaced by

$\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \qquad \text{(E)}\ \text{None of these}$

Dec 13: Which triplet of numbers has a sum NOT equal to 1?

$\text{(A)}\ (1/2,1/3,1/6) \qquad \text{(B)}\ (2,-2,1) \qquad \text{(C)}\ (0.1,0.3,0.6) \qquad \text{(D)}\ (1.1,-2.1,1.0) \qquad \text{(E)}\ (-3/2,-5/2,5)$

Dec 14: Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?

$\text{(A)}\ 15 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 37$

Dec 15: Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period?

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 8$

Dec 16: Casey’s shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job?

$\text{(A)}\ 4 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 12$

Dec 17: Granny Smith has $63. Elberta has$2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have?

$\text{(A)}\ 17 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 23$

Week of Dec 4 – Dec 10: SMC is back!  In preparation for Mathcounts, we will spend a few weeks focusing on probability as it is one of the most challenging topics.  This week we’ll do problems on geometric probability.  The basic formula is

prob[success]  = size or successful region / size of total region

Problems below are inspired from those in Art of Problem Solving, Intro to Probability.

Click GoogleForm to submit answers by Dec 10.
Answers: 2/7; 1/3; (1/2, 1/8, pi/4);  7/8 – See Jason’s solution; 2/7; 5/12;1/6 – See Jason’s solution.

Dec 4: Choose an integer at random from the range (-10,5).  What is the probability that a positive integer is chosen?

Dec 5: Choose a real number at random from the range (-10,5).  What is the probability that a positive number is chosen?  (Hint: use the above formula for geometric probability.)

Dec 6: Two numbers x and y are chosen at random such that $0\le x \le 1$ and $0\le y \le 1$.  What is the probability that (a) $x \le y$? (b) $x+0.5< y$? (c) $x^2+y^2<1$  (Hint: First, draw the region for which $0\le x \le 1$ and $0\le y \le 1$. This is the total region.  Second, draw the region for which (a) is true. This the successful region.  Now use the basic formula to calculate success probability.)

Dec 7: A point is chosen at random inside the square with vertices (0,0), (2,0), (2,2) and (0,2). What is the probability that this point is closer to (0,0) than (3,3)?  (Hint: draw a picture for the total region and the successful region.)

[Jasons’ solution]  The area of the square region within the vertices (0,0), (2,0), (0,2) and (2,2) is 4. Let $L$ be the perpendicular bisector of the segment which connects the points (0,0) and (3,3)It is easy to see that $L$ has slope -1 and intersects the point (3/2, 3/2). Utilizing point slope form, we can deduce the equation of $L$. We have :

$y-{3\over 2} = -(x-{3\over 2}) \rightarrow y=-x+3$

Since $L$ also intersects the square region at (2,1) and (1,2), all points of the enclosed triangular region that is defined by the vertices (2,2), (2,1) and (1,2) is closer to (3,3) than (0,0). The area of this region is 1/2, and the area of the total square region is 4, so the desired probability is (4-1/2) / 4, which is 7/8.

Dec 8: Choose 2 integers $a$ and $b$ at random such that $-3\le a\le 1$ and $-2 \le b \le 4$. What is the probability that the product $ab$ is positive?

Dec 9 Choose 2 real numbers $a$ and $b$ at random such that $-3\le a\le 1$ and $-2 \le b \le 4$. What is the probability that the product $ab$ is positive?

Dec 10: Three points $x$$y$ and $z$ are chosen at random, each from the range (0,1). What is the probability that $x \le y \le z$?

[Jason’s solution] The probability that $x \le y \le z$  should be the same as the probability that $y \le x \le z$ or any other arrangement of the ordering of x, y and z. There are 3! = 6 ways to order x, y and z, so the probability that $x \le y \le z$ is 1/6.