Problem 24 · 2010 Math Kangaroo
Stretch
Number Theory
caseworksum-constraint
Berti’s friends each add together the day and the month of their birthday. They all get the answer 35, but no two of them have the same birthday. What is the largest number of friends Berti can have?
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Answer: B — 8
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Hint 1 of 2
You need months and days with month + day = 35, and each birthday must be a real date.
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Hint 2 of 2
Start from December and step down through the months, checking the day fits that month.
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Approach: count valid (month, day) pairs summing to 35
- List month + day = 35 with a valid day: Dec 23, Nov 24, Oct 25, Sep 26, Aug 27, Jul 28, Jun 29, May 30.
- April would need day 31, which doesn't exist, and earlier months need impossible days.
- That gives 8 different birthdays, so at most 8 friends.
Mark:
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