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Part of 2021 Saint Petersburg Mathematical Olympiad

Problems(3)

Minimum sum of ratios of sides of rectangles

Source: St Petersburg 11.7

A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.

geometryrectangleratio

Reciprocals of pairwise coprime numbers less than two

Source: St Petersburg 2021 10.7

Kolya found several pairwise relatively prime integers, each of which is less than the square of any other. Prove that the sum of reciprocals of these numbers is less than

2

number theoryrelatively prime

Sum of reciprocals of primes in a range is less than 4

Source: St Petersburg 2021 9.7

For a positive integer

n

\sum_{n \le p \le n^4} \frac{1}{p} < 4

where the sum is taken across primes

p

[n, n^4]

Saint Petersburgnumber theory