Bosco

A terminal zero is a $0$ that appears at the end of a number.    

For example, the number $3,800$ has two terminal zeros.    
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?    

          40 • 6 • 75 • 12 = 216,000 ...... so, three terminal zeros    

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The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$    
What is the smallest positive integer that has exactly $2$ perfect square divisors?    

That would be four.  Four has exactly two perfect square divisors, namely 1 and 4.   

Note that the problem does not restrict the divisors to only its two perfect squares.   

We ignore 2 as a divisor of 4, same as the problem ignores 10 as a divisor of 100.   

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Let $x$ and $y$ be complex numbers. If $x + y =2$    

and $xy = 5 - x^2 + xy - y^2$, then what is $x^2 + y^2$?    

ignoring  x + y  =  2    

considering                                 xy  =  5 – x² + xy – y²    

subtract xy from both sides          0  =  5 – x² – y²    

subtract 5 from both sides         – 5  =  – x² – y²    

multiply both sides by – 1              5  =  x² + y²    

                                                      x² + y²  =  5    

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A geometric sequence has 400 terms.    

The first term is 1600 and the common ratio is –1/2.    

How many terms of this sequence are greater than 1000?     

There won't be that many so let's just count them.  

                                                                   1600 • –1/2  =  –800    

                                                                   –800 • –1/2  =    400    

                                                                     400 • –1/2  =  –200    

                                                                   –200 • –1/2  =    100    

                                                                     100 • –1/2  =    –50    

                                                                     –50 • –1/2  =      25    

The numbers are just going to keep dwindling away.     

It's pretty obvious that there is only one term in this      

sequence greater than 1000, i.e., the first one 1600.    

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Find the number of bases $b \ge 2$ such that $100_b - 1_b$ is prime.    

The third position to the left is the base raised to the second power position.    

So this problem could be viewed as "For which bases "b" can (b² – 1) be prime?"    

The only base I could find for which this works is 2 ... i.e., 2² – 1 = 3.    

Therefore, the answer to the problem is:  there is only one.    

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Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.    

The sides are equal and the angles are equal.    

That means the hexagon is a regular hexagon.    

                                                                                               3 sqrt(3)    

The area of a regular hexagon in terms of its sides is   A  =  ––––––––  x  a²  where "a" is a side.    

                                                                                                     2    

                                                                                       A  =  ( 3 x sqrt(3) x 3² ) / 2       

                                                                                       A  =  ( 27 sqrt(3) ) / 2    

Is that good enough or do you need a number.     

                                                                                       A  =  ( 27 x 1.732 ) / 2     

                                                                                       A  =  23.383    

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A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.   
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?    

          40 • 6 • 75 • 12 = 216,000        so, three terminal zeros    

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Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{117}$.    

The 4000^th digit is 5.    

1 / 117 in decimal notation has a repeating phrase. 

1 / 117 = 0.008547     

Let's count 'em up:         0.008547(6^th digit)   008547(12^th digit)   008547(18^th digit)   008547(24^th digit)  • • •       

                                          008547(3990^th digit)   008547(3996^th digit)   0085(4000^th digit)      

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