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.    

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(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)     

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