Particle Moving

A particle moves in a straight line so that its speed is constant for a mile and then changes abruptly for the next mile and is constant for that mile, then changes again, and so on. It moves so that its speed for a given mile varies inversely as the total integral number of miles previously traveled. If the second mile is traversed in 2 hours, how long does it take to travel the nth mile?

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All the Money You Need

If you are tired of feeling broke all the time, here is some algebra that will cheer you up.
Let m be the amount of money that you actually have now,
and let n be the amount of money that you think you need.

A, their average, is given by
A = (m + n) / 2, as usual.

Now, follow this argument and see how much better you feel.
m + n = 2A
Multiply both sides by (mn).
(m + n)(mn) = 2A(mn),
m2n2 = 2Am – 2An.
m2 – 2Am = n2 – 2An,
Now, add A2 to both sides.
m2 – 2Am + A2 = n2 – 2An + A2.
Thus (mA)2 = (nA)2.
So (mA) = (nA), and then m = n.
So you have all the money you need.

Isn’t mathematics wonderful? But how can this be? Can you find a flaw with this reasoning?

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Nasty Numbers

A nasty number is a positive integer having at least four different factors such that the difference between one pair of factors equals the sum of another pair of factors. For instance,

6 is nasty because 6 – 1 = 2 + 3

24 is nasty because 12 – 2 = 4 + 6

30 is nasty because 15 – 2 = 10 + 3.

Find the other six nasty numbers that are less than 180.

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