MathDB

2

Part of 2016 India Regional Mathematical Olympiad

Problems(7)

GCD of terms of Fibonacci style sequence

Source: RMO Delhi 2016, P2

10/11/2016
Consider a sequence (ak)k≥1(a_k)_{k \ge 1} of natural numbers defined as follows: a1=aa_1=a and a2=ba_2=b with a,b>1a,b>1 and gcd⁡(a,b)=1\gcd(a,b)=1 and for all k>0k>0, ak+2=ak+1+aka_{k+2}=a_{k+1}+a_k. Prove that for all natural numbers nn and kk, gcd⁡(an,an+k)<ak2\gcd(a_n,a_{n+k}) <\frac{a_k}{2}.
number theorygreatest common divisor
A bound on product of three numbers

Source: RMO Mumbai 2016, P2

10/11/2016
Let a,b,ca,b,c be positive real numbers such that a1+a+b1+b+c1+c=1.\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1. Prove that abc≤18abc \le \frac{1}{8}.
inequalitiesalgebra
Arranging flags at a summit

Source: RMO Maharashtra and Goa 2016, P2

10/11/2016
At an international event there are 100100 countries participating, each with its own flag. There are 1010 distinct flagpoles at the stadium, labelled 1,#2,...,#10 in a row. In how many ways can all the 100100 flags be hoisted on these 1010 flagpoles, such that for each ii from 11 to 1010, the flagpole #i has at least ii flags? (Note that the vertical order of the flagpoles on each flag is important)
countingcombinatorics
Algebra Inequality

Source: RMO 2016 Karnataka Region P2

10/16/2016
Let a,b,ca,b,c be three distinct positive real numbers such that abc=1abc=1. Prove that a3(a−b)(a−c)+b3(b−c)(b−a)+c3(c−a)(c−b)≥3\dfrac{a^3}{(a-b)(a-c)}+\dfrac{b^3}{(b-c)(b-a)}+\dfrac{c^3}{(c-a)(c-b)} \ge 3
inequalities
Product of three positive reals is less than 1/8

Source: RMO Hyderabad 2016 , P2 .

10/12/2016
Let a,b,ca,b,c be positive real numbers such that a1+a+b1+b+c1+c=1.\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1. Prove that abc≤18abc \le \frac{1}{8}.
inequalitiesalgebrainequalities proposed
RMO 2016 ,Q2

Source: Oct 23,2016

10/25/2016
Let a,b,ca,b,c be positive real numbers such that ab1+bc+bc1+ca+ca1+ab=1\dfrac{ab}{1+bc}+\dfrac{bc}{1+ca}+\dfrac{ca}{1+ab}=1. Prove that 1a3+1b3+1c3≥62\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3} \ge 6\sqrt{2}
inequalities
2016 Chandigarh RMO on stormy night 10 guests came to dinner party no shoes

Source:

8/9/2019
On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there?
maximumalgebracombinatorics