MathDB

10

Part of 2012 AIME Problems

Problems(2)

Sets of perfect squares ending in 256

Source: 2012 AIME I Problem 10

3/16/2012
Let S\mathcal{S} be the set of all perfect squares whose rightmost three digits in base 1010 are 256256. Let T\mathcal{T} be the set of all numbers of the form x−2561000\frac{x-256}{1000}, where xx is in S\mathcal{S}. In other words, T\mathcal{T} is the set of numbers that result when the last three digits of each number in S\mathcal{S} are truncated. Find the remainder when the tenth smallest element of T\mathcal{T} is divided by 10001000.
modular arithmeticAMCAIME
Product of Numbers and Their Floors

Source: 2012 AIME II Problem 10

3/29/2012
Find the number of positive integers nn less than 10001000 for which there exists a positive real number xx such that n=x⌊x⌋n = x \lfloor x \rfloor. Note: ⌊x⌋\lfloor x \rfloor is the greatest integer less than or equal to xx.
floor functionAMC