Problem K. 107. (December 2006)

K. 107. If two digits of the four-digit number 1234 are cancelled in every possible way and the remaining digits are read as a two-digit number, the numbers 12, 13, 14, 23, 24, 34 are obtained. Their sum is 120. Find a four-digit number for which this sum is a) 540; b) 220.

(6 pont)

Deadline expired on January 10, 2007.

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

Megoldás. a) pl. 9876. b) Nincs ilyen. Legyen a keresett négyjegyű szám $\overline{abcd}$ alakú, ekkor a keresett összeg $\overline{ab}+\overline{ac}+\overline{ad}+\overline{bc}+ \overline{bd}+\overline{cd}=10a+b+10a+c+10a+d+10b+c+10b+d+10c+d=30a+21b+12c+3d$ . Ez az összeg osztható 3-mal, a 220 pedig nem. Tehát az ilyen módon előállított számok összege nem lehet 220.

Statistics:

149 students sent a solution.

6 points: 95 students.

5 points: 7 students.

4 points: 12 students.

3 points: 33 students.

1 point: 1 student.

Unfair, not evaluated: 1 solutions.

Problems in Mathematics of KöMaL, December 2006

149 students sent a solution.
6 points:	95 students.
5 points:	7 students.
4 points:	12 students.
3 points:	33 students.
1 point:	1 student.
Unfair, not evaluated:	1 solutions.

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