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4

Part of 1991 Greece National Olympiad

Problems(3)

3 white, 3 green, 3 red 1x1 in 3x3 - 1991 Greece MO Grade X p4

Source:

9/6/2024
In how many ways can we construct a square with dimensions 3×33\times 3 using 33 white, 33 green and 33 red squares of dimensions 1×11\times 1, such that in every horizontal and in every certical line, squares have different colours .
combinatorics
3^x+29=2^y

Source: 1991 Greece MO Grade XI p4

9/6/2024
Find all positive intger solutions of 3x+29=2y3^x+29=2^y.
number theoryDiophantine equationdiophantine
1^{1990}+2^{1990}+3^{1990}+...+1990^{1990} mod 10

Source: 1991 Greece MO Grade XII p4

9/6/2024
If we divide number 11990+21990+31990+...+199019901^{1990}+2^{1990}+3^{1990}+...+1990^{1990} with 1010, what remainder will we find?
number theoryremainder