MHB Find the greatest positive integer

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The discussion focuses on finding the greatest positive integer \( x \) for which the expression \( x^3 + 4x^2 - 15x - 18 \) equals the cube of an integer. Participants highlight that the expression can be bounded by \( (x+1)^3 \), which aids in solving the problem. A breakthrough in the discussion is attributed to a participant's insight, leading to a successful approach. The collaborative nature of the conversation emphasizes problem-solving and sharing strategies. Ultimately, the goal is to determine the maximum value of \( x \) that satisfies the condition.
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Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.
 
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anemone said:
Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.
I had a slice of luck here
x must satisfy

$x^3 + 4x^2 - 15 x - 18 \le (x+1)^3$

or $x^2-18x - 19 \le 0$
so x = 19 makes the RHS 0

so x = 19 is the ans because we get a perfect square ( things would have been different had we not got integer)
 
Well done, kaliprasad!(Yes)

The trick is to recognize that the given expression is less than or equal to $(x+1)^3$.

Thanks for participating as well, my friend!:)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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