Problem B. 3802.

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B. 3802. Given are seven real numbers in such a way that the sum of any three of them is less than the sum of the remaining four. Show that each number is positive.

(3 points)

Deadline expired.

Sorry, the solution is published in Hungarian only.

Megoldás. Legyen a 7 szám $a_1\le a_2\le \ldots \le a_7$ . Ekkor

a₂+a₃+a₄ $\le$ a₅+a₆+a₇.

Ha tehát a₁ $\le$ 0 lenne, akkor

a₁+a₂+a₃+a₄ $\le$ a₅+a₆+a₇

is teljesülne, ami viszont ellentmond a feltételeknek. Ezért hát a₁>0, következésképpen a_i $\ge$ a₁>0 is igaz minden 1 $\le$ i $\le$ 7 esetén.

Statistics on problem B. 3802.

171 students sent a solution.

3 points: 100 students.

2 points: 60 students.

1 point: 7 students.

0 point: 4 students.

Problems in Mathematics of KöMaL, March 2005

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