2018 Adygea Teachers' Geometry Olympiad

Part of Adygea Teachers' Geometry Olympiad

Subcontests

(4)

1

angle between 2 planes, cube (2018 Adygea Teachers' Geometry Olympiad p4)

ABCDA_1B_1C_1D_1

5

BB_1

of the cube , point

K

BK=4

. a) Construct a cube section with the plane

a

passing through the points

K

C_1

parallel to the diagonal

BD_1

. b) Find the angle between the plane

a

BB_1C_1

1

concyclic , intersecting circles (2018 Adygea Teachers' Geometry Olympiad p3)

Two circles intersect at points

A

B

. Through point

B

, a straight line intersects the circles at points

C

D

, and then tangents to the circles are drawn through points

C

D

. Prove that the points

A, D, C

P

- the intersection point of the tangents - lie on the same circle.

1

criterions for right triangles? (2018 Adygea Teachers' Geometry Olympiad p2)

It is known that in a right triangle: a) The height drawn from the top of the right angle is the geometric mean of the projections of the legs on the hypotenuse; b) the leg is the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse. Are the converse statements true? Formulate them and justify the answer. Is it possible to formulate the criterion of a right triangle based on these statements? If possible, then how? If not, why?

1

distances from vertices of square (2018 Adygea Teachers' Geometry Olympiad p1)

Can the distances from a certain point on the plane to the vertices of a certain square be equal to

1, 4, 7

8