# Oct 2018 (AMC8-MC)

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 Oct 28 – Nov 3: Percents.

Oct 28: If 25% of n is 18, then what is 125% of n?

Oct 29: If 20% of a number is 12, then what is 30% of the same number?

Oct 30: Find x if 20% of x equals 12% of (x+20).

Oct 31: Three bags of jelly beans contain 26, 28 and 30 beans. In each bag, the percent of the beans that are yellow is 50%, 25% and 20% respectively. All three bags are dumped into one bowl. What percent of the beans in the bowl are yellow? Round your answer to the nearest integer.

Answer: (26*50%+28*25%+30*20%)/(26+28+30) = (13+7+6)/84 ~ 31%

Nov 1: Cody and Tyler were once the same height. Since then, Tyler has grown 20%, while Cody has grown half as much as Tyler. Cody’s height is now 5 feet 6 inches. How tall is Tyler now?

Answer: Let T be the height of Tyler before he grew 20%. T*1.1=66in. T=60in. Tyler is 72in, or 6ft tall.

Nov 2: Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki can use her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?

Answer: A dozen oranges give 48 ounces of juice, and a dozen pears give 32 ounces. So the percent of pear juice is 32/80=40%.

Nov 3: Grapes are 80% water by weight, and raisins are 20% water by weight. If we start with 500 grams of grapes and remove enough water to turn them into raisins, then what is the weight of the raisins that result?

Answer: Grapes of 500 grams have 100 grams that are not water. Raisins of 125 grams have 100 grams that are not water. So the answer is 125 grams.

Week of Oct 21 – Oct 27: More inequalities.

Oct 21: What values of $x$ satisfy $9+2x-5x \ge -x+12-2x$?

Answer: No value of $x$ works.

Oct 22: What values of $x$ satisfy $2-3x> 6x-3-9x$?

Answer: All values of $x$ work.

Oct 23: Find all positive integers $n$ for which ${1\over 2} < {1\over 3}+{1\over n}$.

Answer: $n = 1, 2, 3, 4, 5$.

Oct 24: What values of $x$ satisfy $7+x \ge 2x+3\ge 12-x$?

Answer: $x = 4$.

Oct 25: What values of $x$ satisfy ${{2x+2}\over {x+7}} >0$?

Answer: $x>-1$ or $x<-7$.

Oct 26: What values of $x$ satisfy ${(x+2)^2\over {x+1}}\le 0$?

Answer: $x<-1$ or $x=-2$.

Oct 27: What values of $x$ satisfy $(x-1)(2-5x)(3x+6)<0$?

Answer: $x>1$ or $-2.

Week of Oct 14 – Oct 20: Inequalities.

Determine whether each of the statements is true or false. If false, provide an example that shows the statement is false.

Oct 14: If $a\le b$ and $b\le c$, then $a.

Answer: False. Counterexample: $a=b=c=2$

Oct 15: If $a\le b\le a$, then $a=b$.

Oct 16: If $a>b$, then $ac > bc$.

Answer: False. Counterexample: $c<0$.

Oct 17: If $x+a\ge y+a$, then $x\ge y$.

Oct 18: If $x\ge y$ and $a\ge b$, then $x+a\ge y+b$.

Oct 19: If $a>b>c>d$, then $ab> cd$.

Answer: False. Counterexample: $a=-1, b=-2, c=-3, c=-4$.

Oct 20: If $a>b>c>d$, then $ac>bd$.

Answer: False. Counterexample: $a=-1, b=-2, c=-3, c=-4$.

Week of Oct 7 – Oct 13: Equations.

Oct 7: If ${2\over 3}={x\over 24}={84\over y}$, what is $x-y$?

Answer: $x=16$, $y=126$, $x-y = -110$.

Oct 8: Solve for $x$: ${1\over 10}+{1\over {10^2}} +{1\over {10^3}} +\dots + {1\over{10^6}} = {x\over {10^6}}$.

Oct 9: Solve for $x$: ${1\over {x-1}} + {5\over 3} = {3\over{x-1}}$

Oct 10: For what number $c$, does the equation $7x+4c=9x+5-2x$ have infinitely many solutions for $x$?

Answer: $c=5/4$

Oct 11: For what numbers $c$, does the equation $7x+4c=9x+5-2x$ have no solutions for $x$?

Answer: $c\ne 5/4$

Oct 12: For what value of $x$, can we not determine the value of ${{x-3} \over {2x+4}}$?

Answer: $x= -2$

Oct 13: A number $x$ is twice its reciprocal. What is $x^6$?

Answer: $x=2/x$. Therefore, $x^6=8$

Week of Sept 30 – Oct 6: All to do with factors.

Sept 30: Without doing the actual division, can you tell which of the following numbers is not divisible by 4?
2884,          201846,         9164,      7632,      79828

Oct 1: Without doing the actual division, can you tell which of the following numbers is not divisible by 8?
7248,          2018640,         100036,     1032,     88056

Oct 2: Without doing the actual division, can you tell which of the following numbers is divisible by 9?

4562,          56233,           12345678,       60231,    9671001

Oct 3: What is the smallest prime factor of $11^7+7^{11}$?

Oct 4: If 18 is a factor of $n$, what other positive numbers must be factors of $n$?
Oct 6: If the four-digit number $72d2$ is divisible by 6, what is the largest possible value of the digit $d$?