11.6

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

cut 10x20 rectangle in 2 pieces III (V Soros Olympiad 1998-99 Round 1 11.6)

Source:

10

x 20

cm rectangle into two pieces with one straight cut so that they can be placed inside the

19.4

cm diameter circle without intersecting.

geometryrectangle

angle chasing, <ABK = <CBL = 2a - 180^o, <ABC=a, angle bisector

Source: V Soros Olympiad 1998-99 Round 2 11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

ABC

B

is obtuse and equal to

a

. The bisectors of angles

A

C

intersect opposite sides at points

P

M

, respectively. On the side

AC

K

L

are taken so that

\angle ABK = \angle CBL = 2a - 180^o

. What is the angle between straight lines

KP

LM

geometryanglesangle bisector

x^x=\frac{1}{\sqrt2}(V Soros Olympiad 1998-99 Round 3 11.6)

Source:

Solve the equation (for positive

x

x^x=\frac{1}{\sqrt2}