10.3

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

(12x-1)(6x-1)(4x-1)(3x -1) = 5 (II Soros Olympiad 1995-96 R1 10.3)

Source:

Solve the equation

(12x-1)(6x-1)(4x-1)(3x -1) = 5.

5 points, 5 arcs, names (II Soros Olympiad 1995-96 R2 10.3)

Source:

A

B

C

D

E

are placed on the circle. In how many ways can the resulting five arcs be designated by the letters

a

b

c

d

e

, if it is forbidden to designate an arc with the same letter as one of its ends? (For example, an arc with ends

A

B

cannot be designated by the letter

a

b

triangle with sides a cosA, b cosB, c cosC

Source: II Soros Olympiad 1995-96 R3 10.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

Each side of an acute triangle is multiplied by the cosine of the opposite angle. a) Prove that a triangle can be formed from the resulting segments. 6) Find the radius of the circle circumscribed around the resulting triangle if the radius of the circle circumscribed around the original triangle is equal to

R

triangle inequalitygeometry