The following problem is from the MATHCOUNTS 2015 State Competition Target Round:¶
When trying to recall some facts about the ages of his three aunts, Josh made the following claims:
- Alice is fifteen years younger than twice Catherine’s age.
- Beatrice is twelve years older than half of Alice’s age.
- Catherine is eight years younger than Beatrice.
The three women’s ages add to exactly one-hundred years. However, Josh’s memory is not perfect, and in fact only three of these four claims are true. If each aunt’s age is an integer number of years, how old is Beatrice?
In [1]:
from sympy import *
a,b,c = symbols('a b c')
In [3]:
claim1=2*c-15-a
claim2=a/2+12-b
claim3=c+8-b
claim4=a+b+c-100
In [4]:
# The combination of claims 1,2,3 results in No Solution
solve([claim1,claim2,claim3],[a,b,c])
Out[4]:
In [5]:
# The combination of claims 1,3,4 results in Non-Integer Solutions
solve([claim1,claim3,claim4],[a,b,c])
Out[5]:
In [6]:
# The combination of claims 1,2,4 results in Non-Integer Solutions
solve([claim1,claim2,claim4],[a,b,c])
Out[6]:
In [7]:
# The combination of claims 2,3,4 results in Integer Solutions. Claim 1 was false.
solve([claim2,claim3,claim4],[a,b,c])
Out[7]:
