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Two 50-term sums and irreducible fractions

Source: Argentine National Olympiad 2014, Level 3, Problem 4

November 6, 2015
algebra

Problem Statement

Consider the following 5050-term sums:
S=11⋅2+13⋅4+...+199⋅100S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100},
T=151⋅100+152⋅99+...+199⋅52+1100⋅51T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}.
Express ST\frac{S}{T} as an irreducible fraction.
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