March 28, 2017

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By Even S.

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We then obtain a - = L pea) p(x) pCb) = ~ ~ b - , L from which it follows that Ip(x) - xlLI < lIn. Since this holds for all n. we see that p(x) = xl L. Conversely, it is easily verified that the function p(x) = xlL actually satisfies (1)-(3). e. the fraction of the total length contained in the interval CD. To return to the original problem. we note that the length of the smaller of the pieces into which the rod is broken will be greater than LI4 if and only if the break point lies within the segment M N (see fig.

Introductory problems 51 multiple of 1000 and the next 999 integers after it, etc. ; hence {O, I, ... , 9}, 1,2, ... , count respectively as the first ten-row, the first hundred-row, etc. In the first ten-row there is exactly one number among whose digits a I occurs, namely the number l. In the first hundred-row, each ten-row except the second (which consists of the numbers from 10 to 19) will contain exactly one number with a I among its digits; the second ten-row consists entirely of numbers with I's among their digits.

What is the probability that they will spell out the word "DEAP'? b. The same process is performed on a set of cards consisting ofthree D's, two O's and one X. What is the probability that the top four cards will spell out the word "DODO"? · Ten slips of paper bearing the numbers 0, I, 2, 3, 4, 5, 6, 7, 8, and 9 are put into a hat. Five slips are drawn at random and laid out in a row in the order in which they were drawn. What is the probability that the five-digit number thus formed will be divisible by 495?

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