Problem C. 1160. (March 2013)
C. 1160. What is the remainder if the sum 20122013+20132012 is divided by 2012.2013?
Suggested by D. Fülöp, Pécs
(5 pont)
Deadline expired on April 10, 2013.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. Vizsgáljuk a következő törtet:
\(\displaystyle t=\frac{2012^{2013}+2013^{2012}}{2012\cdot2013}=\frac{2012^{2012}}{2013}+\frac{2013^{2011}}{2012}.\)
Használjuk fel a következőket:
\(\displaystyle 2012^{2012}=(2013-1)^{2012}=2013^{2012}-2012\cdot2013^{2011}+\binom{2012}{2}\cdot2013^{2010}-...-\binom{2012}{2011}\cdot2013+\binom{2012}{2012}\cdot1=\)
\(\displaystyle =2013a+1,~{\rm{ahol}}~a\in \Bbb N^+;\)
\(\displaystyle 2013^{2011}=(2012+1)^{2011}=2012^{2011}+2011\cdot2012^{2010}+\binom{2011}{2}\cdot2012^{2009}+...+\binom{2011}{2010}\cdot2012+\binom{2011}{2011}\cdot1=\)
\(\displaystyle =2012b+1,~{\rm{ahol}}~b\in \Bbb N^+.\)
Ezekből
\(\displaystyle t=\frac{2013a+1}{2013}+\frac{2012b+1}{2012}=a+\frac{1}{2013}+b+\frac{1}{2012}=a+b+\frac{2012+2013}{2012\cdot2013}=a+b+\frac{4025}{2012\cdot2013}.\)
Vagyis a keresett maradék 4025.
Statistics:
157 students sent a solution. 5 points: 52 students. 4 points: 18 students. 3 points: 37 students. 2 points: 18 students. 1 point: 11 students. 0 point: 20 students. Unfair, not evaluated: 1 solutions.
Problems in Mathematics of KöMaL, March 2013