2019 Adygea Teachers' Geometry Olympiad

Part of Adygea Teachers' Geometry Olympiad

Subcontests

(4)

1

trapezoid conditions (2019 Adygea Teachers' Geometry Olympiad p4)

From which two statements about the trapezoid follows the third: 1) the trapezoid is tangential, 2) the trapezoid is right, 3) its area is equal to the product of the bases?

1

2 balls inside a cube box, radius (2019 Adygea Teachers' Geometry Olympiad p3)

In a cube-shaped box with an edge equal to

5

, there are two balls. The radius of one of the balls is

2

. Find the radius of the other ball if one of the balls touches the base and two side faces of the cube, and the other ball touches the first ball, base and two other side faces of the cube.

1

3 triangles not intersecting in 1 (2019 Adygea Teachers' Geometry Olympiad p2)

Inside the triangle

T

there are three other triangles that do not have common points. Is it true that one can choose such a point inside

T

and draw three rays from it so that the triangle breaks into three parts, in each of which there will be one triangle?

1

3 of 4 areas known, find 4th area (2019 Adygea Teachers' Geometry Olympiad p1)

Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are

S_1, S_2, S_3

. Find the area of the fourth quadrangle.