Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Logic and Speed.
Ed and Sue bike at equal and constant rates and they swim at equal and constant rates. The same way they jog at equal and constant rates. ed covers 74 kms after biking for 2 hrs, jogging for 3 hrs and swimming for 4 hrs while sue covers 91 kms after jogging for 2 hrs swimming for 3 hrs and biking for 4 hrs. Their biking jogging and swimming rates are whole numbers of km/hr, find the sum of the squares of Ed's biking jogging and swimming rates.
Logic
Speed
Integers
Answer: is 314.
AIME I, 2008, Question 3
Elementary Number Theory by David Burton
Let a,b, c be biking jogging and swimming rates then 2a+3b+4c=74 first eqn and 4a+2b+3c=91 second eqn subtracting second from first eqn gives 2a-b-c=17 third eqn
third eqn multiplied by 3 + first eqn gives 8a+c=125 gives \(a \leq 15\) third eqn multiplied by 4 +first eqn gives 10a-b=142 gives \(a \gt 14\)
then a=15 and b=8, c=5 and \(a^{2} +b^{2} + c^{2}\)=225+64+25=314.
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