Mathematical and Physical Journal
for High Schools
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Problem C. 910. (October 2007)

C. 910. Nine integers that add up to 90 are written on the perimeter of a circle. Prove that there are four adjacent numbers that add up to at least 40.

(5 pont)

Deadline expired on November 15, 2007.


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

Megoldás: Legyenek a számok a, b, c, d, e, f, g, h, i - ebben a sorrendben. Bizonyítsunk indirekt: tegyük fel, hogy bármely 4 egymás melletti szám összege kisebb, mint 40. Ekkor:

(a+b+c+d)+(b+c+d+e)+(c+d+e+f)+\ldots+(i+a+b+c)<9\cdot40,

4.(a+b+c+d+e+f+g+h+i)<360,

4.90<360,

ami ellentmondás. Tehát kiinduló feltevésünk nem lehet igaz, vagyis van 4 egymás melletti szám, melyek összege legalább 40.


Statistics:

462 students sent a solution.
5 points:268 students.
4 points:35 students.
3 points:24 students.
2 points:39 students.
1 point:37 students.
0 point:47 students.
Unfair, not evaluated:12 solutions.

Problems in Mathematics of KöMaL, October 2007