Problems of the Week

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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.

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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