10.10

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(2)

each deputy quarreled with exactly 3 other deputies

Source: II Soros Olympiad 1995-96 R1 10.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

Each deputy of the Academic Duma quarreled with exactly three other deputies. The President ordered the Speaker to divide the deputies into n factions so that agreement reigned within one faction. For what smallest

n

is this always possible? (This means that there is such

n

that deputies could always be divided into

n

factions, but not always into

(n- 1)

min area, semicircle and triangle related

Source: II Soros Olympiad 1995-96 R3 10.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

The Order "For Faithful Service" of the

7

th degree in shape is a combination of a semicircle with a diameter

AB = 2

AM B

AM

BM

intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?

geometrygeometric inequalityareas