MathDB

11.6

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

max no of obtuse triangles by 16 segments (III Soros Olympiad 1996-97 R1 11.6)

Source:

5/29/2024
What is the largest number of obtuse triangles that can be composed of 1616 different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?
geometrycombinatoricscombinatorial geometry
199 corrupted ministers

Source: III Soros Olympiad 1996-97 R2 11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
In one criminal kingdom, an underdeveloped state, the King decided to start a fight against corruption and, as an example, punish one of his 199199 ministers. The ministers were summoned to the palace and seated at a large round table. At first they wanted to find the one who had the most money in his bank account and declare him the main corrupt official. It takes 2020 minutes to determine the amount of money in the bank account of one minister. But the King ordered that the accused be found within four hours while he underwent medical procedures. According to the Noble Court Administrator, any minister can be accused, you just need to find a legal justification.The Chief Lawyer proposed that the first minister discovered, who has more money in his bank account than each of his two neighbors (one on the right and one on the left), be declared corrupt. How can one be sure to find a minister who meets this condition within the allotted 44 hours? (During this time, it is possible to consistently determine the size of the bank accounts of no more than 1212 ministers. It is assumed that the amount of money in bank accounts is different.)
combinatorics
log_{x+y} (x^2+y^2)<= 1 (III Soros Olympiad 1996-97 R3 11.6)

Source:

5/31/2024
On the coordinate plane, draw a set of points M(x,y)M(x,y), the coordinates of which satisfy the inequality log⁡x+y(x2+y2)≤1.\log_{x+y} (x^2+y^2) \le 1.
logarithmsinequalitiesanalytic geometry