1) Find the smallest perfect square whose decimal representation begins and ends with three 4's.
2) Consider the number whose digits consist
of the decimal digits of the numbers from 2000 to 1 in descending
Find the first three prime factors of this number. (The first can be found relatively easily.)
3) The game of Yahtzee® is played with five six-sided dice. A "Yahtzee" is when all five dice show the same face. You begin by rolling all five dice. You are allowed two subsequent rolls in which you can roll any subset of the five dice. Assuming that your only goal is to get a Yahtzee and that you play optimally, what is the probability of succeeding?
4) At twelve noon and twelve midnight, the hour, minute, and second hands of a clock are all aligned. At what time between 1 PM and 11 PM are the three hands as close together as possible? [Note: By this I mean that maximum angle between the hands is as small as possible.]
5) How many ways are there to factor 1000000 into three distinct positive integers if the order of the factors is irrelevant?
6) What are the last four digits of
where "1998" appears 1998 times?
[Recall that iterated exponents are computed from "top to bottom". For example:
is equal to 3^(3^3)=3^27.]
7) There are 2^n "words" of length n using an alphabet consisting of the letters a and b. For example, the words of length 2 are aa, ab, ba, and bb. It is easy to find a word of length 5 so that, reading from left to right, every two letter word appears; reading aabba yields aa, ab, bb, and ba.
Can you find a word of length 19 so that, reading from left to right, every four letter word appears?
8) The integer 30 can be written as a sum
of three consecutive positive integers:
30 = 9 + 10 + 11; moreover, it can be written as a sum of more than one consecutive positive integer in exactly three ways, namely
30 = 9 + 10 + 11 = 6 + 7 + 8 + 9 = 4 + 5 + 6 + 7 + 8.
Find an integer which can be written as a sum of 10 consecutive positive integers and which can be written as a sum of more than one consecutive positive integer in exactly 10 ways.
9) If X + Y = 1 and X^2 + Y^2 = 2, what is X^3 + Y^3 ?
10) If X + Y + Z = 1, X^2 + Y^2 + Z^2 = 2, and X^3 + Y^3 + Z^3 = 3, what is X^4 + Y^4 + Z^4?
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