Bosco

There is a group of five children, where two of the children are twins. In how many ways can I distribute $3$ identical pieces of candy to the children, if the twins must get an equal amount of candy?     

First, give the twins each one piece of candy.       

That leaves three kids and one piece of candy.     

There are 3 ways you could give that last piece.    

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In quadrilateral PQRS, PQ = QR = RS = SP, PR = 4, and QS = 16.  Find the perimeter of PQRS.    

There's a formula for the perimeter of a rhombus in terms of its diagonals.    

                              P  =  2 • sqrt(p² + q²)   where p and q are the diagonals      

                              P  =  2 • sqrt(16 + 256)  =  2 • sqrt(272)  =  2 • sqrt(16 • 17)       

                              P  =  8 • sqrt(17)    

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The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

     Median = (Base₁ + Base₂) / 2     

     Median = (12 + (12+16) ) / 2  =  40 / 2    

     Median = 20    

     The area of the trapezoid is irrelevant to this problem.    

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Find the number of ways that Magnus can give out $12$ identical stickers to $2$ of his friends, if every friend gets at least one sticker.    

Friend 1  Friend 2    

     1           11    

     2           10    

     3             9    

     4             8    

     5             7    

     6             6    

     7             5    

     8             4    

     9             3    

   10             2    

   11             1    

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1    

                                                  (x²) / (x-1+x³)   

At x = 1, it equals 1 / 1.  As x gets larger than 1, the denominator increases faster than the numerator.    

That's a condition that causes the fraction to get smaller and smaller, but it never quite reaches zero.    

So, the fraction approaches zero.  It approaches zero but never gets there.  It's called an asymptote.    

I think they call this approaching zero asymptotically.    

_.    

(a) Find the last two digits of $9*5*3$.    

(b) Find the last two digits of $1^{2^3}$.    

     (By convention, exponent towers are evaluated from the top down, so $1^{2^3} = 1^{(2^3)}$.)    

(a)  3 & 5.  9 • 5 • 3 = 135.   

(b)  No answer.  1^(2^3) doesn't have two last digits.  1 raised to any power is still 1.    

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When the same constant is added to the numbers 60, 120, and 160, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?    

In a geometric progression the ratio between succeeding numbers is the same.   

Therefore                                            120 + x         160 + x    

                                                            ––––––   =   ––––––     

                                                             60 + x          120 + x     

Cross multiplying                                 (x + 120)(x + 120)  =  (x + 60)(x + 160)     

                                                             x² + 240x +14,400  =  x² + 220x + 9600       

Subtract x² from both sides                  240x + 14,400  =  220x + 9600    

Subtract 220x from both sides                20x + 14,400  =  9600   

Subtract 10,000 from both sides                            20x  =  – 4800       

                                                                                   x  =  – 240    

Plugging – 240  

into the first ratio                                        – 120   

                                                                 –––––––  =  0.6666 • • •       

                                                                   – 180   

Plugging – 240 

into the second ratio                                   – 80  

                                                                 –––––––  =  0.6666 • • •     

                                                                   – 120              

The common ratio, after     

adding the constant, is                               0.6666 • • •    (or  ²/₃ if you prefer)     

_.    

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    

_.    

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.   

_.    

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    

_.    

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.    

_.   

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.    

_.   

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    

_.   

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    

_.   

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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