MathDB

9

Part of 2010 Malaysia National Olympiad

Problems(3)

OMK 2010

Source: Malaysia National Olympiad 2010 Bongsu Category Problem 9

6/4/2011
A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?
algebra
OMK 2010

Source: Malaysia National Olympiad 2010 Muda Category Problem 9

6/4/2011
Let mm and nn be positive integers such that 2n+3m2^n+3^m is divisible by 55. Prove that 2m+3n2^m+3^n is divisible by 55.
Eulermodular arithmeticnumber theory unsolvednumber theory
OMK 2010

Source: Malaysia National Olympiad 2010 Sulung Category Problem 9

6/4/2011
Show that there exist integers mm and nn such that mn=50+73−50−73.\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.
number theory unsolvednumber theory