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3

Part of 2012 India Regional Mathematical Olympiad

Problems(3)

Given a+b=1; show expression<=1(Well-known)

Source: RMO 2012

12/2/2012
Let aa and bb be positive real numbers such that a+b=1a+b=1. Prove that aabb+abba≤1a^ab^b+a^bb^a\le 1.
logarithmsfunctioninequalitiesinequalities unsolved
2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}

Source: CRMO 2012 region 5 p3 Mumbai

9/30/2018
Solve for real xx : 22x⋅23{x}=11⋅25{x}+5⋅22[x]2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}
(For a real number x,[x]x, [x] denotes the greatest integer less than or equal to x. For instance, [2.5]=2[2.5] = 2, [−3.1]=−4[-3.1] = -4, [π]=3[\pi ] = 3. For a real number x,{x}x, \{x\} is defined as x−[x]x - [x].)
algebranumber theoryfractional partsolve- integer partInteger Partsolve- fractional part
(2^x-1)(2^y-1)=2^{2^z}+1[Indian RMO 2012(b) Q3]

Source:

12/2/2012
Find all natural numbers x,y,zx,y,z such that (2x−1)(2y−1)=22z+1.(2^x-1)(2^y-1)=2^{2^z}+1.
modular arithmetic