MathDB

8

Part of 2010 Malaysia National Olympiad

Problems(3)

OMK 2010

Source: Malaysia National Olympiad 2010 Bongsu Category Problem 8

6/4/2011
Find the last digit of 71×72×73×⋯×72009×72010.7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.
modular arithmeticnumber theory unsolvednumber theory
OMK 2010

Source: Malaysia National Olympiad 2010 Muda Category Problem 8

6/4/2011
For any number xx, let ⌊x⌋\lfloor x\rfloor denotes the greatest integer less than or equal to xx. A sequence a1,a2,⋯a_1,a_2,\cdots is given, where an=⌊2n+12⌋.a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor. How many values of kk are there such that ak=2010a_k=2010?
floor functionnumber theory unsolvednumber theory
OMK 2010

Source: Malaysia National Olympiad 2010 Sulung Category Problem 8

6/4/2011
Show that log⁡abc+log⁡bca+log⁡cab≥4(log⁡abc+log⁡bca+log⁡cab)\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b) for all a,b,ca,b,c greater than 1.
logarithmsinequalitiesinequalities unsolved