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?

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

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 $\displaystyle 5 (a + b + c + d + e) = 28 \Longrightarrow a + b + c + d + e = 5.6$ .

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

Problem

Solution