An Interesting Problem on Inequalities and Number Theory

Introduction

The following problem appeared in the IOQM 2023 paper.

There is a brute force solution for this, where we either cross-multiply by β to get,

and then take cases by increasing the value of β from 1,2,3... till we get a possible integral value of α satisfying the above inequality; or flip the entire inequality by taking reciprocal, then cross-multiplying by α instead and repeat a similarly case work on values of α till we get a possible integral solution of β.

Rather than resorting to computational brute force, we can employ elementary number theory to derive an optimal solution. The key lies in recognizing the underlying structure of the problem.

This solution is not designed for the time-constrained environment of an exam. Instead, it offers a detailed and insightful exploration using fundamental tools, breaking down each step to deepen understanding and provide a fresh perspective on the problem.

Analysis of the Problem from a Number Theory Perspective

Our first step would be no different than the one we discussed in the brute force case. We cross-multiply by β to get,

Now we simply get rid of all fractions in the following manner,

This gives us a very nice way to represent the value of 592α. We can write that

Now we can easily see that GCD (592,256) = 16 from above product form of the numbers. Using this information, we get that,

Now we use a very basic yet important result from number theory to find the complete solution set for the above reduced equation. As GCD (37,16) = 1, from Bezout's Theorem we get that, there exists integers x, y such that,

We can use Division Algorithm to find the required x and y as follows,

Now the important thing to notice is that Bezout's Theorem only gives us existence of such x and y, not uniqueness. There is an infinitely family of solutions to that equation. Let us try to find that family of solutions. Let x, y be a solution. Then we have,

Now using the fact that GCD (37,16) = 1, we get that 16 completely divides x+3 and 37 completely divides 7-y. Hence to make the equality true we must have that,

Hence the general solution for that equation is,

Now we can use this family of solutions to get a family of solutions for the equation,

We simply multiply on both sides of our previous family of solutions by p,

Remark: The family of solutions presented above, is a subset of the complete solution set to the equation

While this family satisfies the equation, it does not exhaust all possible solutions.

Let us now find the complete set of solutions for the above equation. Let x, y be a solution. Then we get that,

Using same logic as before we get that,

This is the general family of solutions of that equation. Now we want solution to the equation,

Comparing coefficients we get,

We need to find minimum value of β. Now let us recall that β is a positive integer. Also, p is a positive integer. So, we need to make sure that ap + b is positive, if not then β will become negative.

Notice another thing. We can make the term ap + b equal to any positive integer. We just need to take a = 0 and let b be any positive integer. So, fix a = 0. Then for β we get,

Now the most obvious step would be to put the value of β in the above inequality.

Using this inequality we get,

Now we know b is any positive integer, and to get minimum value in above expression we need b to be minimum too. So, minimum occurs at b = 1. Hence,

Well, it may seem like we are done but we still need to check the validity of our solution by checking whether the corresponding α is a positive integer or not.

Hence our answer is valid.

Conclusion

This problem required to use few basic concepts in number theory over and over till we were left with a trivial expression. In many such problems which we tend to use case bashing can be solved like this in a more elegant manner.