MathDB

6.1 Three shooters - Anilov, Borisov and Vorobiev - made

6

each shots at one target and scored equal points. It is known that Anilov scored

43

points in the first three shots, and Borisov scored

43

points with the first shot knocked out 3 points. How many points did each shooter score per shot? if there was one hit in 50, two in 25, three in 20, three in 10, two in 5, in 3 - two, in 2 - two, in 1 - three? https://cdn.artofproblemsolving.com/attachments/a/1/4abb71f7bccc0b9d2e22066ec17c31ef139d6a.png
6.2 / 7.4 Prove that a

10 \times 10

chessboard cannot be covered with

25

figures like https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png.
6.3 The squares of a chessboard contain natural numbers such that each is equal to the arithmetic mean of its neighbors. Sum of numbers standing in the corners of the board is

16

. Find the number standing on the field

e2

.
6.4 There is a table

100 \times 100

. What is the smallest number of letters which can be arranged in its cells so that no two are identical the letters weren't next to each other?
6.5 The pioneer detachment is lined up in a rectangle. In each rank the tallest is noted, and from these pioneers the most short. In each row, the lowest one is noted, and from them is selected the tallest. Which of these two pioneers is taller? (This means that the two pioneers indicated are the highest of the low and the lowest of tall - must be different)
6.6 Find the product of three numbers whose sum is equal to the sum of their squares, equal to the sum of their cubes and equal to

1

.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here.

Problem Statement