Thursday, January 17, 2013

2013 - 33 &34

Most of the time, I hate questions that include the words "How many different ways  ...", but this one wasn't bad. The other one? Wasn't to be this year.  Didn't have the slightest clue.

Difficulty: Hard, until you ask the question correctly. Then it's medium.

Difficulty: No clue.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013

1. Both great questions:
For #34 consider breaking the equation into the four possible intervals:

When x+y>=0 and x-y>=0 (or y>=-x and y<=x) we have: x+y+x-y=10. x=5 and -5<=y=<5

When x+y>=0 and x-y<=0 (or x>=-y and x<=y) we have: x+y-x+y=10. y=5 and -5<=x=<5

When x+y<=0 and x-y>=0 (or x<=-y and x>=y) we have: -x-y+x-y=10. y=-5 and -5<=x=<5

When x+y<=0 and x-y<=0 (or y<=-x and y>=x) we have: -x-y-x+y=10. x=-5 and -5<=y=<5

The expression implies we want the largest value of |x| and largest value of –y. We should pick x=+-5 and y=-5. This gives us 25+25+40 or 90.

2. Yup, that makes sense.