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?

Note: "Exact-change combination" doesn't necessarily mean that you must use more than 1 kind of coins listed above. For example, paying 4 quarters is allowed.

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 ABC.\overline{{\color{#D61F06}A}{\color{#3D99F6}B}{\color{#20A900}C}}.

I give you the following sum:

ACBBACBCACAB+CBA1223\begin{array}{rcccc} && {\color{#D61F06}A} & {\color{#20A900}C} & {\color{#3D99F6}B} \\ && {\color{#3D99F6}B} & {\color{#D61F06}A} & {\color{#20A900}C} \\ && {\color{#3D99F6}B} & {\color{#20A900}C} & {\color{#D61F06}A} \\ && {\color{#20A900}C} & {\color{#D61F06}A} & {\color{#3D99F6}B} \\ + && {\color{#20A900}C} & {\color{#3D99F6}B} & {\color{#D61F06}A} \\ \hline &1 & 2 & 2 & 3 \end{array}

What is ABC?\overline{{\color{#D61F06}A}{\color{#3D99F6}B}{\color{#20A900}C}}?

Note: The digits A,{\color{#D61F06}A}, B,{\color{#3D99F6}B}, and C{\color{#20A900}C} are not necessarily distinct.

p(x)=x7+ax5+bx3+cx 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, r = 1,2, and 3,3, what is the greatest integer k k such that p(n) p(n) is divisible by k k for all integers n? 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 ab,\frac{a}{b}, where aa and bb are coprime positive integers.

What is a+b?a+b?


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