0% Complete
0/19 Steps

A sequence of natural numbers and a recurrence relation

Understand the problem

Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 = 0$, $ a_1 = 1$ and
\[ a_n = 2a_{n - 1} + a_{n - 2},\]
for $ n\geq2.$

$ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides
$ a_{m + j} + ( - 1)^ja_{m - j}$.

$ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$

Source of the problem

Indian National Mathematical Olympiad 2010

Topic
Number Theory
Difficulty Level
Medium
Suggested Book
Problem Solving Strategies by Arthur Engel

Start with hints

Do you really need a hint? Try it first!

For (a), use strong induction on j.
The recurrence has the characteristic equation x^2-2x-1=0 (see this post for details). This has two distinct roots, 1\pm\sqrt{2}. Hence the general solution is of the form y_n=a(1+\sqrt{2})^n+b(1-\sqrt{2})^n.
Solving for a,b, we get \[a_n=\frac{(1+\sqrt{2})^n-(1-\sqrt{2})^n}{2\sqrt{2}}\]. Suppose that 2^k\;|\; n, i.e. n=2^kj for some j. Now it suffices to show that \frac{a_n}{2^k}= \[\frac{(1+\sqrt{2})^{2^kj}-(1-\sqrt{2})^{2^kj}}{2^{k+1}\sqrt{2}}\] is an integer. Use induction on k to prove this.
Note that the given expression is equivalent to \frac{(1+\sqrt{2})^{2^{k-1}j}+(1-\sqrt{2})^{2^{k-1}j}}{2}\cdot \frac{(1+\sqrt{2})^{2^{k-1}j}-(1-\sqrt{2})^{2^{k-1}j}}{2^k\sqrt{2}}. By induction, it suffices to show that \frac{(1+\sqrt{2})^{2^{k-1}j}+(1-\sqrt{2})^{2^{k-1}j}}{2} is an integer. Write b_n=\frac{(1+\sqrt{2})^{n}+(1-\sqrt{2})^{n}}{2}. Note that b_n satisfies b_n=2b_{n-1}+b_{n-2} hence it suffices to prove that b_0 and b_1 are both integers. As b_0=1=b_1, we are done. 

Watch the video (Coming Soon)

Connected Program at Cheenta

Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Similar Problems

Solving a congruence

Understand the problemProve that the number of ordered triples   in the set of residues of $latex p$ such that , where  and  is prime is . Brazilian Olympiad Revenge 2010 Number Theory Medium Elementary Number Theory by David Burton Start with hintsDo you really need...

Inequality involving sides of a triangle

Understand the problemLet  be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet  be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...

Vectors of prime length

Understand the problemGiven a prime number  and let  be  distinct vectors of length  with integer coordinates in an  Cartesian coordinate system. Suppose that for any , there exists an integer  such that all three coordinates of  is divisible by . Prove that .Kürschák...

Missing digits of 34!

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...

An inequality involving unknown polynomials

Understand the problemFind all the polynomials  of a degree  with real non-negative coefficients such that  , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

Hidden triangular inequality (PRMO Problem 23, 2019)

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

PRMO – 2019 – Questions, Discussions, Hints, Solutions

This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you. 1. 42. 133. 134. 725. 106. 297. 518. 499. 1410. 5511. 612. 1813. 1014. 5315. 4516. 4017. 3018. 2019. 1320. Bonus21. 1722....

Bangladesh MO 2019 Problem 1 – Number Theory

A basic and beautiful application of Numebr Theory and Modular Arithmetic to the Bangladesh MO 2019 Problem 1.

Functional equation dependent on a constant

Understand the problemFind all real numbers  for which there exists a non-constant function  satisfying the following two equations for all i)  andii)  Baltic Way 2016 Functional Equations Easy Functional Equations by BJ Venkatachala Start with hintsDo you really need...

Pigeonhole principle exercise

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...