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University of South Carolina High School Math Contest/1993 Exam/Problem 7

Problem

Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?

Usc93.7.PNG
$\mathrm{(A) \ }4.2 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }5.6 \qquad \mathrm{(D) \ }6.2 \qquad \mathrm{(E) \ }6.8$

Solution

If we call the squares $a,b,c,d,e,f$ (in order from left to right), we have: $b+c+d+e+f=3$, $a+c+d+e+f=8$, $a+b+d+e+f=5$, $a+b+c+e+f=6$, $a+b+c+d+f=2$, $a+b+c+d+e=4$. Adding all the equations gives us $5 (a + b + c + d + e) = 28 \Longrightarrow a + b + c + d + e = 5.6$.

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