Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 561. (November 2017)

K. 561. A novel was published in three volumes. The page numbering started with 1 in the first volume, and in the second and third volumes it continued where the previous volume ended. The second volume was 50 pages thicker than the first one, and the third volume was 1.5 times as thick as the second volume. The sum of the first page numbers in the three volumes was 893. How many pages long is the entire novel? How many digits were used altogether in numbering the pages?

(6 pont)

Deadline expired on December 11, 2017.

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

Megoldás. Ha \(\displaystyle x\) oldalas egy-egy könyv, akkor az első oldalszámok rendre \(\displaystyle 1\), \(\displaystyle x+1\) és \(\displaystyle 2x+51\). Tehát \(\displaystyle 3x+53 = 893\), innen rendezéssel az \(\displaystyle x=280\)-at kapjuk. Tehát az első könyv \(\displaystyle 280\), a második \(\displaystyle 330\), a harmadik pedig \(\displaystyle 1,5\cdot330=495\), így a regény \(\displaystyle 1105\) oldalas. A számjegyek száma:

\(\displaystyle 9\cdot1+(99-10+1)\cdot2+(999-100+1)\cdot3+(1105-1000+1)\cdot4=\)

\(\displaystyle =9\cdot1+90\cdot2+900\cdot3+106\cdot4=9+180+2700+424=3313.\)


144 students sent a solution.
6 points:75 students.
5 points:9 students.
4 points:6 students.
3 points:44 students.
2 points:7 students.
1 point:2 students.
0 point:1 student.

Problems in Mathematics of KöMaL, November 2017