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2018-11-26 Intermediate

I want to buy a $1.00 newspaper. I have just enough pennies ($0.01), nickels ($0.05), dimes ($0.10), and quarters (\$0.25) to buy the newspaper in every possible exact-change combination of those coins.

How many coins do I have?

You have 8 batteries, and you know 4 are "good" and the other 4 are "bad."

You have a device that takes 6 batteries, but only 4 need to be "good" for it to work. You can test your device by picking 6 of the batteries and trying to turn the device on, but otherwise there is no way of checking if the batteries are good.

In the worst-case scenario, how many tests do you need so you know of a particular set of batteries which will get your device working?

The device accepts 6 batteries, although only 4 need to be working. Image via TED-Ed and Artrake Studio.

I am thinking of a 3-digit number $$\overline{{\color{red}A}{\color{blue}B}{\color{green}C}}.$$

I give you the following sum:

$\begin{array}{rcccc} && {\color{red}A} & {\color{green}C} & {\color{blue}B} \\ && {\color{blue}B} & {\color{red}A} & {\color{green}C} \\ && {\color{blue}B} & {\color{green}C} & {\color{red}A} \\ && {\color{green}C} & {\color{red}A} & {\color{blue}B} \\ + && {\color{green}C} & {\color{blue}B} & {\color{red}A} \\ \hline &1 & 2 & 2 & 3 \end{array}$

What is $$\overline{{\color{red}A}{\color{blue}B}{\color{green}C}}?$$

Note: The digits $${\color{red}A},$$ $${\color{blue}B},$$ and $${\color{green}C}$$ are not necessarily distinct.

$p(x) = x^7 + ax^5 + bx^3 + cx$

Given that all of this polynomial's 7 roots are real and three of them are $$r = 1,2,$$ and $$3,$$ what is the greatest integer $$k$$ such that $$p(n)$$ is divisible by $$k$$ for all integers $$n?$$

Four isosceles right triangles with legs of length 3 are inside a circle of radius 5 and positioned on a unit grid.

Consider every possible chord of the circle that does not touch a triangle. Then shade every portion of the white inside the circle that is not touched by any of such chords.

The shaded area is $$\frac{a}{b},$$ where $$a$$ and $$b$$ are coprime positive integers.

What is $$a+b?$$

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