Analytical problem from CSMO 2018

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(China Southeast Mathematical Olympiad 2018 Grade 11 P8)
Given a positive real $C geq 1$ and a sequence $a_1, a_2, ...$ of nonnegative real numbers satisfy
$$left|xlnx-sum_k=1^[x] left[fracxkright]a_kright| leq Cx,$$
where $[x]$ is the floor function of $x.$
Prove that for any real $y geq1,$
$$sum_k=1^[y] a_k<3Cy.$$
An obvious step might be this well-known identity:
$$sum_k=1^[x] left[fracxkright]a_k=sum_k=1^[x] Fleft(fracxkright),$$
where $F(x)$ is the sum of all $a_k, kleq x.$
Because this is olympiad-level, there should be an elementary solution but perhaps the bound could be improved using higher mathematics?




number-theory elementary-number-theory analytic-number-theory




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    (China Southeast Mathematical Olympiad 2018 Grade 11 P8)
    Given a positive real $C geq 1$ and a sequence $a_1, a_2, ...$ of nonnegative real numbers satisfy
    $$left|xlnx-sum_k=1^[x] left[fracxkright]a_kright| leq Cx,$$
    where $[x]$ is the floor function of $x.$
    Prove that for any real $y geq1,$
    $$sum_k=1^[y] a_k<3Cy.$$
    An obvious step might be this well-known identity:
    $$sum_k=1^[x] left[fracxkright]a_k=sum_k=1^[x] Fleft(fracxkright),$$
    where $F(x)$ is the sum of all $a_k, kleq x.$
    Because this is olympiad-level, there should be an elementary solution but perhaps the bound could be improved using higher mathematics?




    number-theory elementary-number-theory analytic-number-theory




    share|cite|improve this question





















      up vote
      2
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      favorite









      up vote
      2
      down vote

      favorite











      This has been recently posted on aops:



      (China Southeast Mathematical Olympiad 2018 Grade 11 P8)
      Given a positive real $C geq 1$ and a sequence $a_1, a_2, ...$ of nonnegative real numbers satisfy
      $$left|xlnx-sum_k=1^[x] left[fracxkright]a_kright| leq Cx,$$
      where $[x]$ is the floor function of $x.$
      Prove that for any real $y geq1,$
      $$sum_k=1^[y] a_k<3Cy.$$
      An obvious step might be this well-known identity:
      $$sum_k=1^[x] left[fracxkright]a_k=sum_k=1^[x] Fleft(fracxkright),$$
      where $F(x)$ is the sum of all $a_k, kleq x.$
      Because this is olympiad-level, there should be an elementary solution but perhaps the bound could be improved using higher mathematics?




      number-theory elementary-number-theory analytic-number-theory




      share|cite|improve this question











      This has been recently posted on aops:



      (China Southeast Mathematical Olympiad 2018 Grade 11 P8)
      Given a positive real $C geq 1$ and a sequence $a_1, a_2, ...$ of nonnegative real numbers satisfy
      $$left|xlnx-sum_k=1^[x] left[fracxkright]a_kright| leq Cx,$$
      where $[x]$ is the floor function of $x.$
      Prove that for any real $y geq1,$
      $$sum_k=1^[y] a_k<3Cy.$$
      An obvious step might be this well-known identity:
      $$sum_k=1^[x] left[fracxkright]a_k=sum_k=1^[x] Fleft(fracxkright),$$
      where $F(x)$ is the sum of all $a_k, kleq x.$
      Because this is olympiad-level, there should be an elementary solution but perhaps the bound could be improved using higher mathematics?





      number-theory elementary-number-theory analytic-number-theory





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