9.7
Problems(2)
300 people in a circle, last person standing wins
Source: II Soros Olympiad 1995-96 R1 9.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
6/2/2024
people took part in the drawing for the main prize of the television lottery. They lined up in a circle, then, starting with someone who received number , they began to count them. Moreover, every third person dropped out every time. (So, in the first round, everyone with numbers divisible by dropped out). The counting continued until there was only one person left. (It is clear that more than one circle was made). This person received the main prize. (It “accidentally” turned out to be the TV director’s mother-in-law). What number did this person have in the initial lineup?
combinatoricsnumber theory
min of largst areas by 2 straight lines through a point of triangle with area 1
Source: II Soros Olympiad 1995-96 R3 9.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
6/4/2024
Through a point located on a side of a triangle of area , two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let be the largest of the areas of these parts. Find the smallest possible value of .
geometryareasgeometric inequality