# Dec 2017 (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.

This week we are having a logic stretch with a variety of puzzles.  I hope you are having fun during your week off from school.  Happy holidays to all!

Week of Dec 24 – Jan 30

Dec 24:  Five friends are standing in a line. Cliff is standing directly behind Danny, and there are two people between Cliff and Mark. Mark is standing somewhere behind Eric, but somewhere in front of Tom. Which of the five friends is fourth in line?

Answer: Danny  (Cliff, Danny, Tom, Mark, Eric)

Dec 25:  Today Darron’s teacher pairs each student with a partner to create exactly 12 pairs of students. Next week each student will be paired with a different partner. Darron’s partner for next week only can be chosen from how many students?  (Merry Christmas to those who celebrate!)

Answer: 22 (24 students in total)

Dec 26: Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number
of pencils he could have left over?

Dec 27:  In the land of Noggin Knockers, the inhabitants greet each other by bumping heads. At a certain gathering a total of 36 bumps were exchanged. If each person there bumped heads exactly once with each other person, how many people were at the gathering?

Answer: 9 people  (x(x-1)/2 = 36, x=9)

Dec 28: If I have two more brothers than sisters and each of my brothers also has two more brothers than sisters, how many more brothers than sisters does my oldest sister have?

Dec 29:  The heights of six students Joe, Mary, Sue, Steve, Lisa and John are 60 inches, 64 inches, 58 inches, 68 inches, 63 inches and 69 inches. Sue is 4 inches shorter than Joe. The girls are the three shortest students. Steve is 1 inch shorter than John. Mary is the
shortest student. What is the sum of John’s height and Lisa’s height, in inches?

Answer: 132 (Mary 58, Sue 60, Lisa 63, Joe 64, Steve 68, John 69)

Dec 30: There are 64 identical-looking coins, one of which is slightly heavier than the others. A balance scale can be used to show which one of two groups of coins is heavier or that the two groups weigh the same. What is the minimum number of uses of the balance scale that is guaranteed to determine which of the coins is the heavier one?

Answer: 4 times. First weighing: divide 64 coins into 3 groups, 21-21-22. Weigh group 1 (21 coins) against group 2 (21 coins).  If either one is heavier, it contains the heavy coin; else group 3 (22 coins) has the heavy one.   Second weighing: assume the 22-coin group contains the heavy one.  divide it into 3 groups 7-7-8. Weigh group 1 (7 coins) against group 2 (7 coins). Again we know which group has the heavy coin. Third weighing assume the 8-coin group contains the heavy one.  Divide into 3-3-2. Weigh group 1 (3 coins) against group 2 (3 coins).  Fourth weighing finds the coin.

Last week, we had a problem on numbers in a base different from 10.  Many of you had a little bit of trouble.  In fact, problems converting between different bases are not hard since there is a formula to do it.  I’m attaching this link that explains how to do it.

Click GoogleForm to submit answers by Sat, Dec 23. (I’m posting answers on Jan 7,  so you have more time to enjoy your holiday week.)

Week of Dec 17 – Dec 23

Dec 17: $2_{10} = ( ~~ )_2$, $4_{10} = (~~ )_2$$16_{10} = ( ~~ )_2$$64_{10} = ( ~~ )_2$

Dec 18: $3_{10} = ( ~~ )_2$, $7_{10} = ( ~~ )_2$$15_{10} = ( ~~ )_2$$63 _{10} = ( ~~ )_2$

Dec 19: $234_{10} = ( ~~ )_2$, $87_{10} = ( ~~ )_2$, $321_{10} = ( ~~ )_2$

Dec 20: $1011_2 = ( ~~ )_{10}$, $110_2 = ( ~~ )_{10}$, $10101_2 = ( ~~ )_{10}$

Dec 21: $120_3 = ( ~~)_{10}$, $120_{10} = ( ~~ )_3$

Dec 22: $32_4 = ( ~~ )_{10}$, $32_{10} = ( ~~ )_4$

Dec 23: $100_5 = ( ~~ )_{10}$, $100_{10} = ( ~~ )_5$

We will practice Mathcounts problems till the end of Feb.

Week of Dec 10 – Dec 16

Dec 10: If 11× n = 123,456,787,654,321, what is the value of n?

Dec 11: The point (4, −2) is reflected over the line y = x. What are the coordinates of its

Dec 12: A chess club has 8 girls and 6 boys. Two members, Zig and Zag, are fraternal twins of different genders. If a team of 3 girls and 3 boys is randomly selected for the district championship, what is the probability that exactly one of the twins is on the team? Express your answer as a common fraction.

Answer: 1/2  (The total number of possible ways to choose the team is (8 choose 3)x(6 choose 3).  The desired number of ways is (7 choose 2)x(5 choose 3) + (7 choose 3)x(5 choose 2).

Dec 13: If a, b and c satisfy the equations $a^2 + b^2= 313$, $b^2 + c^2 = 277$ and $a^2 + c^2 = 302$, what is the value of  $a^2 + b^2 - c^2$ ?

Answer: 180  ($2c^2=277+302-313=266$.

Dec 14: What is the value of $32 _4+ 43_5 + 54_6$ when written in base 7? ($32_4$ is 32 written in base 4)

Answer: $131_7$.  ($32_4=14$, $43_5=23$, $54_6=34$. $14+23+34=71=131_7$)

Dec 15: The ages, in years, of four members of a family are represented by a, b, c and d, where a < b < c < d. Their mean age is 34, their median age is 33, and the range of their ages is 32. What is the value of a?

Answer: 19  (The 4 numbers are 19, x, 66-x, 51.)

Dec 16: Raquel uses six different digits to fill in the blanks below, writing one digit in
each blank, so that the resulting addition statement is correct. What is the least
possible sum of the six digits?   __ + __ = __

This week’s 7 problems come from last year’s school round, which is when each school selects its team.

Week of Dec 3 – Dec 9

Dec 3: Samhir writes down all of the odd numbers between 500 and 700 that are divisible by both 7 and 9. What is the sum of the numbers Samhir writes?

Dec 4: Sasha’s secret passcode is a nine-digit number that begins and ends with 6. The sum of every three consecutive digits in the number is 14. What is the fifth digit of Sasha’s passcode?

Answer: 2  (1st, 4th, 7th and 9th are 6. Therefore, the 8th is 2, and so is the 5th.)

Dec 5: If p is prime and n is even such that p + n = 47 and pn = 210, what is the value of n?

Answer: n = 42, p = 5

Dec 6: If $a \# b = a^2$ $(7-b)$, what is the value of $(2 \# 5) \# 3$?

Dec 7: If the permutations of the letters in the word SURE are numbered 1 through 24 in alphabetical order, what number is RUSE?

Answer: 12  (Here only 4 letters, E, R, S and U are permuted.  RUSE is the last word beginning of R.  There are 6 words beginning with E and 6 beginning with R, so RUSE is 12th.)

Dec 8: A fair 10-sided die, with faces numbered 1 through 10, is rolled once. What is the probability that the number rolled will be prime? Express your answer as a common fraction.

Answer: 2/5  (Four primes: 2, 3, 5, 7)

Dec 9: What is the area of the triangle enclosed by the lines y = 0, x = 8 and y = x?

Answer: 32  (Triangle with base 8 and height 8)