Bosco

Find the smallest positive integer B so that when we express the decimal number 301 as a base B number, we still get a 2-digit number.   

Well, in base 10 it's a three digit number, so the base has to be bigger than ten.  

Per https://www.rapidtables.com/convert/number/base-converter.html it's base 18 ... 301₁₀ = GD₁₉    

For information, the next one down was 17 ... 310₁₀ = 10C₁₇    

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            Prove that given any set of $17$ integers, there exist nine of them whose sum is divisible by $2.$    

            How about demonstrating an exception?  A set of seventeen 3's.  You did say "any" set.  

            3 3 3   3 3 3   3 3 3    

            3 3 3   3 3 3   3 3    

            Choose and add up any nine of them.  The total is 27.  Not divisible by 2.  

               Real numbers a and b satisfy  
               a + ab = 250  
               a - ab = -240  
               Enter all possible values of a, separated by commas.     

               Really?  Just add them together.....   

               a + ab  =    250    

               a – ab  =  –240    

               2a  =  10    

                 a  =  5    

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                                                  ab⁴ = 48           (eq 1)    

                                                  ab² = 4             (eq 2)    

solve (eq 2) for a                       a = 4 / b²     

sub a back in to (eq 1)               (4 / b²) • b⁴  =  48    

                                                   (b⁴ / b²)  =  48 / 4    

                                                             b²  =  12   

                                                               b  =  2 • sqrt(3)    

The problem doesn't ask for a     

 but sub b² back in to (eq 2)             a • 12  =  4    

                                                                a  =  4 / 12    

                                                                a  =  1 / 3     

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All members of our painting team paint at the same rate. If $10$ members can paint a $1000$ square foot wall in $10$ minutes, then how long would it take for $5$ members to paint a $1000$ square foot wall, in minutes?     

Same size wall, but half as many painters.  It will take twice the time.  20 minutes     

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In quadrilateral $PQRS$, $PQ = QR = RS = SP$, $PR = \sqrt{2}$, and $QS = \sqrt{2}$. Find the perimeter of $PQRS$.   

All four sides are equal; that makes it at least a rhombus.  

Both diagonals are the same; and that makes it a square.   

The diagonal of the square is the square root of two. 

That makes the sides equal one.  Therefore the perimeter is 4 units.    

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AX² + CX²  =  BX² + DX²   

3² + 3²  =  3² + DX²   

DX²  =  3² + 3² – 3²    

DX²  =  9     ====>>          DX = +3    

discard negative value      DX = 3    

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