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Algebra

Arithmetic Progressions

Arithmetic Progressions: Level 3 Challenges

         

1000 chocolates are placed on a table. On their covers, they are labelled \(1,2,3,4, \cdots ,1000\). Two friends Sandeep and Abhiram eat chocolates labelled in the arithmetic progressions \(3,6,9,12,15,\cdots\) and \(7,14,21,28,35,\cdots\) respectively. If ever there is any conflict between them i.e. if both are entitled to eat the same chocolate, Sandeep eats it. So, what is the total number of chocolates which are not eaten?

Let \(x^3+ax^2+bx+c\) be a polynomial with positive integer coefficients which are in arithmetic progression in that order.

Determine the maximum possible number of integer roots of the polynomial.

What is the largest positive integer that must be a divisor of the sum of an arithmetic sequence with 90 integer terms?


Try Part II

\[ \begin{array}{llllll} A_1 : & 2, & 9, & 16, \ldots , & 2 + (1000-1) \times 7 \\ A_2: & 3, & 12 , & 21, \ldots, & 3 + (1000-1) \times 9? \\ \end{array} \]

How many integers appear in both of the following arithmetic progressions above?

Details and assumptions

Since 2 appears in \(A_1\) but not in \(A_2\), it does not appear in both of the arithmetic progressions.

Given \(p\) arithmetic progressions, each of which consisting of \(n\) terms, if their first terms are \(1,2,3,\ldots,p-1,p\) and common differences are \(1,3,5,7,\ldots,2p-3,2p-1,\) respectively, what is the sum of all the terms of all the arithmetic progressions?

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