Q) Show that there cannot be distinct positive integers such that and . Solution: This problem trivially arrives to a scenario that we have covered in the module ‘System of Equations’. And it touches a very important concept that is useful in many scenarios. We have . Cubing both sides, and cancelling out the common terms (assuming… Continue reading
Category: Polynomials and Equations
INMO – 1986 | A Problem on Polynomials
Q) If is a polynomial with integer coefficients and three distinct integers, then show that it is impossible to have Solution: The solution revolves around an important property of integer coefficient Polynomials, For any 2 integers , is always divisible by . So from this we have divides which implies divides , divides , divides . Once we have this, the remaining… Continue reading
Week -4 | System of Equations – 7
This is an optional module for the students preparing for engineering entrance examinations Q-1) Let be a positive integer such that is prime. Choose for , such that the are not all equal. Also let be a polynomial such that for all . Show that the degree of is at least . Q-2) Let be distinct integers.… Continue reading
Week -4 | System of Equations – 6
Practice Problems (MCQ Questions may have more than 1 correct answer) Q-1) Solve the equation in reals . Q-2) If , be the roots of the equation , then the value of is a) b) c) d) None of these Q-3) Let be such that has no real roots. Show that the… Continue reading
Week -4 | System of Equations – 5
This is an optional module for the students preparing for engineering entrance examinations Q-1) Find the number of natural numbers which satisfy the following 2 conditions: a) b) divides Solution: , this is divisible by for any natural number 2. Note that for any number to be divisible by , it must be divisible by its… Continue reading