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2018-12-03 Intermediate

         

If Earth somehow contracted to \(\frac {1}{8 }\) of its current volume instantly (through some magical internal forces), while keeping its shape and mass, then how long would a day be?

Three unit circles are drawn tangent to each other with all their centers lying on line segment \(OE.\)

From \(O,\) line segment \(OD\) is drawn such that it's tangent to the rightmost circle.

If the length of the chord \(AB\) is \(\frac ab\), where \(a\) and \(b\) are coprime positive integers, then what is \(a+b?\)


Note: I did not create this problem; I simply solved it and decided to post it on here. Credit goes to my math teacher.

After paying for a newspaper with 4 quarters, the newspaper came out. But, to my surprise, my 4 quarters also came back out because the machine was broken. I quickly took note of how I paid for the paper: "4 quarters." After spending over a quarter of an hour at the machine, I paid for a newspaper in every possible way.

How many newspapers did I end up with?

Assumptions:

  • Only 4 kinds of coins can be used: pennies ($.01), nickels ($.05), dimes ($.10), and quarters ($.25).
  • The order of the coins placed into the machine doesn't matter.

The 3 little pigs have recently established their own construction company.

  1. The eldest pig has a daily wage of $350 and builds a house in 9 days.
  2. The middle pig has a daily wage of $250 and builds a house in 12 days.
  3. The youngest pig has a daily wage of $150 and builds a house in 16 days.

Their new client is the wolf, now a real estate entrepreneur. The wolf would like to contract the pigs to build a house. The pigs provide the following conditions on their work:

  • Exactly 2 pigs work on the house each day (so that the other pig can call for help if something happens).
  • Each pig must have at least 1 day off during this working period.
  • If the job is done before the end of the last day, the wolf is still charged a full-day rate for that day.

Under the 3 pigs' conditions, what is the least amount of money (in $) the wolf can invest to complete this project?

The \(x\)-axis is a common tangent of the two curves \( y = x^{2} \) and \( y = x^{3} \).

Suppose the slope of the other common tangent (in sky blue) of these two curves is expressed as \( \frac{p}{q},\) where \( p \) and \( q \) are coprime positive integers.

What is \( p+q?\)

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