Problem C. 1210. (February 2014)

C. 1210. We have four sacks of flour. By three measurements, the following information is obtained: the first and second sacks together are lighter than the other two sacks, the first and third together have the same mass as the other two, and the first and fourth together are heavier than the other two. Which sack is the heaviest?

(5 pont)

Deadline expired on March 10, 2014.

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

Megoldás. Legyen az első zsák tömege \(\displaystyle a\) kg, a másodiké \(\displaystyle b\) kg, a harmadiké \(\displaystyle c\) kg, a negyediké \(\displaystyle d\) kg. Ekkor \(\displaystyle a+b<c+d\) (1), \(\displaystyle a+c=b+d\) (2) és \(\displaystyle b+c<a+d\) (3).

(1) és (2) összege: \(\displaystyle 2a+b+c<b+c+2d\), amiből \(\displaystyle 2a<2d\), vagyis \(\displaystyle a<d\) következik.

(1) és (3) összege: \(\displaystyle a+2b+c<a+c+2d\), innen \(\displaystyle b<d\) következik.

Végül (2) és (3) összege: \(\displaystyle a+b+2c<a+b+2d\), amiből \(\displaystyle c<d\).

Tehát \(\displaystyle d\)-nél \(\displaystyle a\), \(\displaystyle b\) és \(\displaystyle c\) is kisebb, vagyis a negyedik zsák a legnehezebb.

Megjegyzés: Ilyen számok léteznek, pl. \(\displaystyle a=3\), \(\displaystyle b=1\), \(\displaystyle c=2\) és \(\displaystyle d=4\).

Statistics:

168 students sent a solution.
5 points:	135 students.
4 points:	11 students.
3 points:	9 students.
2 points:	1 student.
1 point:	2 students.
0 point:	1 student.
Unfair, not evaluated:	9 solutionss.

Problems in Mathematics of KöMaL, February 2014