Select Page

# Understand the problem

Find all integer solutions $(a,b)$ of the equation$\$(a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\$$

##### Source of the problem
Costa Rica NMO 2010
Number Theory
6/10
##### Suggested Book
A Friendly Introduction to Number Theory by J.H.Silverman

Do you really need a hint? Try it first!

Write the equation in this form : $(a-b)^2 = 3(ab-1)(ab+2a+2b+3)$ The idea is that the RHS is a 4 degree polynomial in two variables and the LHS is a 2 degree polynomial in the same two variables. Now, you have the intuition that the 4th-degree polynomial will surpass the LHS after some time. We, therefore, aim to bound the a and b assuming the equality holds.

So, we try to find when RHS $\geq$ LHS. Rather we will find the condition when $(a-b)^2 \leq (ab-1)(ab + 2a + 2b + 3)$. Assume $a \geq b$. We will find when $latex (a-b) \leq (ab-1)$ and $(a-b) \leq (ab + 2a + 2b + 3)$.

$latex (a-b) \leq (ab-1)$ – For this to hold. $(a^2-1)(b^2 -1) \geq 0$. $(a-b) \leq (ab + 2a + 2b + 3)$ – For this to hold $(a+3)(b+1) \geq 0$.

Hence, try to understand that we have essentially bounded the solutions. Observe this implies that to solutions to have essentially. $latex -3 \leq a,b \leq 1$. Hence show that by computation that : The solution set is $\boxed{(-3,-3), (-3,0), (0,-3), (-2,1), (1,-2), (-1,-1), (1,1)}$

# 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

## Geometry of AM GM Inequality

AM GM Inequality has a geometric interpretation. Watch the video discussion on it and try some hint problems to sharpen your skills.

## Geometry of Cauchy Schwarz Inequality

Cauchy Schwarz Inequality is a powerful tool in Algebra. However it also has a geometric meaning. We provide video and problem sequence to explore that.

## RMO 2019 Maharashtra and Goa Problem 2 Geometry

Understand the problemGiven a circle $latex \Gamma$, let $latex P$ be a point in its interior, and let $latex l$ be a line passing through $latex P$. Construct with proof using a ruler and compass, all circles which pass through $latex P$, are tangent to \$latex...

## RMO 2019 (Maharashtra Goa) Adding GCDs

Can you add GCDs? This problem from RMO 2019 (Maharashtra region) has a beautiful solution. We also give some bonus questions for you to try.

## Number Theory, Ireland MO 2018, Problem 9

This problem in number theory is an elegant applications of the ideas of quadratic and cubic residues of a number. Try with our sequential hints.

## Number Theory, France IMO TST 2012, Problem 3

This problem is an advanced number theory problem using the ideas of lifting the exponents. Try with our sequential hints.

## Algebra, Austria MO 2016, Problem 4

This algebra problem is an elegant application of culminating the ideas of polynomials to give a simple proof of an inequality. Try with our sequential hints.

## Number Theory, Cyprus IMO TST 2018, Problem 1

This problem is a beautiful and simple application of the ideas of inequality and bounds in number theory. Try with our sequential hints.

## Number Theory, South Africa 2019, Problem 6

This problem in number theory is an elegant applciations of the modulo technique used in the diophantine equations. Try with our sequential hints

## Number Theory, Korea Junior MO 2015, Problem 7

This problem in number theory is an elegant application of the ideas of the proof of infinitude of primes from Korea. Try with our sequential hints.