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 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

Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.

Rectangular Piece of Paper | AMC 10A, 2014| Problem No 22

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2014. Problem-23. You may use sequential hints to solve the problem.

Probability in Marbles | AMC 10A, 2010| Problem No 23

Try this beautiful Problem on Probability from AMC 10A, 2010. Problem-23. You may use sequential hints to solve the problem.

Points on a circle | AMC 10A, 2010| Problem No 22

Try this beautiful Problem on Number theory based on Triangle and Circle from AMC 10A, 2010. Problem-22. You may use sequential hints to solve the problem.

Circle and Equilateral Triangle | AMC 10A, 2017| Problem No 22

Try this beautiful Problem on Triangle and Circle from AMC 10A, 2017. Problem-22. You may use sequential hints to solve the problem.