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Problems Calculus Number sequences Boundedness, monotonicity Algebra Polynomials Algebraic identities for polynomials Identical transformations Sequences

14–16 5.0

Prove that for any natural number $a_{1} > 1$ there exists an increasing sequence of natural numbers $a_{1}, a_{2}, a_{3}, \dots$ , for which $a_{1}^{2} + a_{2}^{2} + \dots + a_{k}^{2}$ is divisible by $a_{1} + a_{2} + \dots + a_{k}$ for all $k \geq 1$ .

Problem #PRU-105072

Problems Algebra Polynomials Algebraic identities for polynomials Identical transformations

12–14 2.0

Two different numbers $x$ and $y$ (not necessarily integers) are such that $x^{2} - 2000 x = y^{2} - 2000 y$ . Find the sum of $x$ and $y$ .