Question

In a magic triangle, each of the six whole numbers $10$ through $15$ is placed in one of the circles so that the sum $S$ of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is

$36$

$37$

$38$

$39$

$40$

Answer: D

Concepts:magic square optimization

Difficulty rating: 1140

Small Hint:

Adding the three side sums counts each corner number twice and each midpoint number once

Big Hint:

Let $C$ be the sum of the corner numbers. Then $3S = 75 + C;$ put the three largest numbers at the corners

Solution:

Let $C$ be the sum of the corner numbers. Adding the three side sums gives $3S = 75 + C,$ because the six numbers sum to $75$ and every corner is counted one extra time. To maximize $S,$ place the three largest numbers $13,$ $14,$ $15$ at the corners, giving $C = 42.$

Then $3S = 75 + 42 = 117,$ so $S = 39.$ This is achievable: put $12$ between corners $13$ and $14,$ put $10$ between $14$ and $15,$ and put $11$ between $15$ and $13.$

Thus, the correct answer is D .

1985 AMC 8 Problem 24

Answer: D

Difficulty rating: 1140

Small Hint:

Big Hint:

Solution:

Problem 24 in Other Years