Understand the problem

Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms.

Source of the problem
Bulgarian Mathematical Olympiad 2002
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!

There exists a general theory of second-order linear difference equations. Read about it here.
Can we have a_n>a_{n+1} for all n? What happens if we do? What happens otherwise?

Note that if a_n<a_{n+1} for some n then the sequence becomes increasing thereafter. Use this fact to simplify the recurrence relation and solve it explicitly.

The sequence cannot be decreasing because it is a sequence of positive integers. Hence there exists (a smallest) k such that a_k\le a_{k+1}. If a_k=a_{k+1} then the sequence becomes constant from the kth term onwards (we shall treat this case later). Otherwise 3a_{k+1}-2a_k>a_k hence a_{k+2}>a_{k+1}. This implies that the sequence becomes increasing from the kth term onwards. Also, a_{n+2}=3a_{n+1}-2a_n for n\ge k. This difference equation has the characteristic equation \lambda^2-3\lambda +2=0 (see the link in hint 1) which has the solutions \lambda = 2,1. Thus, a_{n+k}=2^nA+B for A,B satisfying A+B=a_k, 2A+B=a_{k+1}. Take any prime divisor p of A+B. By Fermat’s little theorem, 2^{m(p-1)} \equiv 1 \; (\text{mod}\; p) for every positive integer m. Thus 2^{m(p-1)}A+B\equiv A+B\equiv 0\; (\text{mod}\; p). Hence the sequence contains infinitely many composites. This cannot be allowed, so the sequence cannot be strictly increasing at any point.  

The above discussion shows that, for any permissible sequence, there exists a (smallest) j and a prime q such that a_n=q for all n\ge j. For n<j, the sequence is decreasing. Note that, either q=a_{j+1}=3a_j-2a_{j-1}=3q-2a_{j-1} or q =2a_{j-1}-3q. Hence, either a_{j-1}=q or a_{j-1}=2q. The first one can happen only if j=1 because otherwise the minimality of j is violated. In that case, the sequence is constant and b=c=q. If a_{j-1}=2q then either j=2 (in which case b=2q, c=q) or q=|6q-2a_{j-2}| hence a_{j-2}=3q\pm\frac{q}{2}. The last equality forces q to be 2. Thus a_{j-2}=6\pm 1. If j>3 then 4=a_{j-1}=|18\pm 3 - 2a_{j-3}| which is absurd as 2a_{j-3} cannot be an odd number. Hence j=3 in this case and c=4,b=5,7.

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...