MathDB

7

Part of 2007 AIME Problems

Problems(2)

Summation, Ceiling, and Floor Functions

Source: AIME I 2007 #7

3/15/2007
Let N=∑k=11000k(⌈log⁡2k⌉−⌊log⁡2k⌋).N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor). Find the remainder when N is divided by 1000. (Here ⌊x⌋\lfloor x \rfloor denotes the greatest integer that is less than or equal to x, and ⌈x⌉\lceil x \rceil denotes the least integer that is greater than or equal to x.)
functionceiling functionlogarithmsfloor functioncalculusintegrationLaTeX
Floor of cube roots

Source: AIME II 2007 #7

3/29/2007
Given a real number x,x, let ⌊x⌋\lfloor x \rfloor denote the greatest integer less than or equal to x.x. For a certain integer k,k, there are exactly 7070 positive integers n1,n2,…,n70n_{1}, n_{2}, \ldots, n_{70} such that k=⌊n13⌋=⌊n23⌋=⋯=⌊n703⌋k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor and kk divides nin_{i} for all ii such that 1≤i≤70.1 \leq i \leq 70. Find the maximum value of nik\frac{n_{i}}{k} for 1≤i≤70.1\leq i \leq 70.
geometry3D geometryfloor functioncalculusintegrationAMC