# Oct 2018 (AMC8-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 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.

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

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

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 $ab> cd$.

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

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: $x\ne -2$

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 $n$?

Answer: 1, 2, 3, 6, 9

Oct 5: What is the units digit of the product of any five consecutive positive integers?

Oct 6: If the four-digit number $72d2$ is divisible by 6, what is the largest possible value of the digit $d$?