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A group of 200 high school students, 105 girls and 95 boys, is randomly divided into two rows of 100 students each. Each student in one row is directly opposite a student in the other row, and all of the opposite pairs shake hands. Prove that the number of “girl-girl” handshakes is five more than the number of “boy-boy” handshakes. |
Solution 1 by Michael Kilgore, Grand Haven High School
Let the number of boy-boy handshakes be represented by B, the number
of girl-girl handshakes be represented by G, and number of boy-girl
handshakes be represented by M. Then we have 95 = 2B +
M and 105=2G + M. This is true because we have two boys
in each boy-boy hand-shake, two girls in each girl-girl handshake, and one boy
(or girl) in each boy-girl handshake. We then solve the equation, 95 - 2B
=105 - 2G, and arrive at G = B + 5, which means that
the number of girl-girl handshakes is five greater than the number of boy-boy
handshakes.
Solution 2 by Lisa Tian, Plymouth-Canton Educational Park
Let x be the number of boy-girl handshakes. Then
Also solved by: David Zhang (Ann Arbor Huron H.S.); Long Dinh Trinh (Charlotte H.S.); Arnav Moudgil (Grosse Pointe North H.S.); Adam Carlson, Aaron Kauffman, Dana Leenheer, Elise Miller (Grand Rapids Forest Hills Northern H.S.); Myron Chang, Vikram Raghunathan (Plymouth-Canton Educational Park).
