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Part of 2008 Tuymaada Olympiad

Problems(2)

Portraits of famous scientists hang on a wall

Source: Tuymaada 2008, Junior League, First Day, Problem 1.

Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly

5\over 4

times greater than Vasya's result. What minimum number of portraits can be on the wall? Author: V. Frank

combinatorics unsolvedcombinatorics

Several irrational numbers are written on a blackboard

Source: Tuymaada 2008, Senior League, First Day, Problem 1.

Several irrational numbers are written on a blackboard. It is known that for every two numbers

a

b

on the blackboard, at least one of the numbers a\over b\plus{}1 and b\over a\plus{}1 is rational. What maximum number of irrational numbers can be on the blackboard? Author: Alexander Golovanov

combinatorics unsolvedcombinatorics