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Problem 726

AIME late

Combinatorics Difficulty 5.3 Find the answer

Problem 5.3. In five of the nine circles in the picture, the numbers 1, 2, 3, 4, 5 are written. Replace the digits $6, 7, 8, 9$ in the remaining circles $A, B, C, D$ so that the sums of the four numbers along each of the three sides of the triangle are the same.

Construct the correspondence

- In circle $A$
- In circle $B$
- In circle $C$
- In circle $D$
- the number 6 is placed.
- the number 7 is placed.
- the number 8 is placed.
- the number 9 is placed.

Your answer

Official solution

Answer: $A=6, B=8, C=7, D=9$ .

Solution. From the condition, it follows that $A+C+3+4=5+D+2+4$ , from which $D+4=A+C$ . Note that $13 \geqslant D+4=A+C \geqslant 6+7$ . Therefore, this is only possible when $D=9$ , and $A$ and $C$ are 6 and 7 in some order. Hence, $B=8$ .

The sum of the numbers along each side is $5+9+3+4=20$ . Since $5+1+8+A=20$ , then $A=6$ . Since $6+C+3+4=20$ , then $C=7$ .

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