Competition problem — Combinatorics

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Combinatorics Difficulty 5.3 AIME, harder Find the answer

16. As shown in Figure $5, A, B, \cdots, F$ represent six different numbers 1,2 , $\cdots, 6$ . Each of the five lines passes through some of these points. By adding the numbers corresponding to the points on each line, five sums can be obtained, and the sum of these five numbers is 47. Then the number corresponding to point $B$ is:

Pick one

Solution

16. E.
From the problem, we know that these five sums are $A+B+C$ ,
$A+E+F, \ C+D+E, \ B+D, \ B+F \text{.}$

Adding them together, we get
$\begin{array}{l} 2 A+3 B+2 C+2 D+2 E+2 F=47 \\ \Rightarrow 2(A+B+C+D+E+F)+B=47. \end{array}$

Since $A$ to $F$ are six different numbers from 1 to 6, then
$A+B+C+D+E+F=1+2+\cdots+6=21 \text{.}$

Therefore, $B=47-21 \times 2=5$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.