MathDB

4.

Part of 2020 Saint Petersburg Mathematical Olympiad

Problems(3)

2/3*6 + 2*5/3*6*9 + ... + 2*5*...*2015/3*6...*2019

Source: Saint Petersburg MO 2020 Grade 11 Problem 4

5/7/2020
The sum 23⋅6+2⋅53⋅6⋅9+…+2⋅5⋅…⋅20153⋅6⋅9⋅…⋅2019\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019} is written as a decimal number. Find the first digit after the decimal point.
number theory
Inequality over integers

Source: Saint Petersburg MO 2020 Grade 10 Problem 4

5/7/2020
Let mm be a given positive integer. Prove that there exists a positive integer kk such that it holds 1≤1m+2m+3m+…+(k−1)mkm<2.1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.
inequalitiesnumber theory
Game of Columns

Source: Saint Petersburg MO 2020 Grade 9 Problem 4

5/8/2020
On a table with 2525 columns and 300300 rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at least a marked square will be deleted. Kostya wants to be left as few as possible columns in the table, and Lesha wants there to be as many as possible columns in the table. What is the largest number of columns Lesha can guarantee to leave?
combinatorics