**Problem: **Let be a quadratic polynomial where are real numbers. Suppose be an arithmetic progression of positive integers. Prove that are integers.

**Discussion: **

.

According to the problem

are in arithmetic progression of positive integers.

Clearly

This implies

implying b is negative.

Now we know is an integer.

Then (replacing )

This implies is an integer. But is also an integer. Hence is an integer.

Now we also know is an integer.

Then (replacing )

Again replacing we get is an integer or is an integer.

Note that is some positive integer. Let it be . Then where c is some positive integer (as we know b is negative)

or

(suppose). Then or

squaring both sides we get

Right hand side is rational. Hence left hand side is also rational. This implies is rational. Since c is an integer, this implies is integer. Hence b is integer.

We know . Since b is integer, therefore is integer.

Again is integer and is integer, implies a must be rational.

Finally, if a is rational and is integer then a must be integer.

## Chatuspathi:

**Paper:**RMO 2015 (West Bengal)**What is this topic:**Polynomial**What are some of the associated concepts:**Rational and surd**Where can learn these topics:**Cheenta**Book Suggestions:**Polynomials by Barbeau

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