Mathematical and Physical Journal
for High Schools
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Problem B. 3802. (March 2005)

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 pont)

Deadline expired on April 15, 2005.


Sorry, the solution is available only in Hungarian. Google translation

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

a2+a3+a4\lea5+a6+a7.

Ha tehát a1\le0 lenne, akkor

a1+a2+a3+a4\lea5+a6+a7

is teljesülne, ami viszont ellentmond a feltételeknek. Ezért hát a1>0, következésképpen ai\gea1>0 is igaz minden 1\lei\le7 esetén.


Statistics:

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