MathDB

8

Part of 2018 South East Mathematical Olympiad

Problems(2)

Exponent of five in a certain expression

Source: 2018 China Southeast MO Grade 10 P8

7/31/2018
Given a positive integer mm. Let Al=(4l+1)(4l+2)...(4(5m+1)l)A_l = (4l+1)(4l+2)...(4(5^m+1)l) for any positive integer ll. Prove that there exist infinite number of positive integer ll which 55ml∣Al and 55ml+1∤Al5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l and find the minimum value of ll satisfying the above condition.
number theory
Pvove sum less than 3Cy

Source: 2018 China Southeast MO Grade 11 P8

7/31/2018
Given a positive real C≥1C \geq 1 and a sequence a1,a2,a3,⋯a_1, a_2, a_3, \cdots satisfying for any positive integer n,n, an≥0a_n \geq 0 and for any real x≥1x \geq 1, ∣xlg⁡x−∑k=1[x][xk]ak∣≤Cx,\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx, where [x][x] is defined as the largest integer that does not exceed xx. Prove that for any real y≥1y \geq 1, ∑k=1[y]ak<3Cy.\sum_{k=1}^{[y]}a_k < 3Cy.
algebra