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INMO 2013 Question No. 3 Solution

3     Let \((a,b,c,d \in \mathbb{N})\) such that \((a \ge b \ge c \ge d)\). Show that the equation \((x^4 – ax^3 – bx^2 – cx -d = 0)\) has no integer solution.

Sketch of the Solution:

Claim 1: There cannot be a negative integer solution. Suppose other wise. If possible x= -k (k positive) be a solution.

Then we have \((k^4 + ak^3 +ck = bk^2 +d)\). Clearly this is impossible as \((a\ge b , k^3 \ge k^2 )\) and \((c \ge d )\).

Claim 2: 0 is not a solution (why?)

Claim 3: There cannot be a positive integer solution. Suppose other wise. If possible x=k (k positive) be a solution.

Then we have \((k^4 = a k^3 + b k^2 + c k + d)\)
This implies that the right hand side is divisible by k which again implies that d is divisible by k (why?).
Let d=d’k
Now \((c\ge d) \implies (c \ge d’k) \implies (c \ge k)\).
Thus \((a \ge c \ge k ) \implies (a \cdot k^3 \ge k \cdot k^3 )\).
Hence the equality \((k^4 = a k^3 + b k^2 + c k + d)\) is impossible.

February 8, 2013

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