MathDB

given side equality, find angle

Source: Moldova 2000 Grade 7 P8

4/23/2021

Points

D

and

N

on the sides

AB

and

BC

and points

E,M

on the side

AC

of an equilateral triangle

ABC

, respectively, with

E

between

A

and

M

, satisfy

AD+AE=CN+CM=BD+BN+EM

. Determine the angle between the lines

DM

and

EN

.

geometryTriangle

parallelepiped, given relation between dimensions

Source: Moldova 2000 Grade 8 P8

4/25/2021

A rectangular parallelepiped has dimensions

a,b,c

that satisfy the relation

3a+4b+10c=500

, and the length of the main diagonal

20\sqrt5

. Find the volume and the total area of the surface of the parallelepiped.

geometry3D geometry

operations on 2000 numbers

Source: Moldova 2000 Grade 9 P8

4/26/2021

Initially the number

2000

is written down. The following operation is repeatedly performed: the sum of the

10

-th powers of the last number's digits is written down. Prove that in the infinite sequence thus obtained, some two numbers will be equal.

game

circular geometry configuration

Source: Moldova 2000 Grade 10 P8

4/26/2021

Two circles intersect at

M

and

N

. A line through

M

meets the circles at

A

and

B

, with

M

between

A

and

B

. Let

C

and

D

be the midpoints of the arcs

AN

and

BN

not containing

M

, respectively, and

K

and

L

be the midpoints of

AB

and

CD

, respectively. Prove that

CL=KL

.

geometry

lines intersect on circumcircle

Source: Moldova 2000 Grade 11 P8

4/27/2021

In an isosceles triangle

ABC

with

BC=AC

and

\angle B<60^\circ

,

I

is the incenter and

O

the circumcenter. The circle with center

E

that passes through

A,O

and

I

intersects the circumcircle of

\triangle ABC

again at point

D

. Prove that the lines

DE

and

CO

intersect on the circumcircle of

ABC

.

geometrycircumcircle

geo ineq with inradii

Source: Moldova 2000 Grade 12 P8

4/28/2021

A circle with radius

r

touches the sides

AB,BC,CD,DA

of a convex quadrilateral

ABCD

at

E,F,G,H

, respectively. The inradii of the triangles

EBF,FCG,GDH,HAE

are equal to

r_1,r_2,r_3,r_4

. Prove that

r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.

geometrygeometric inequality

Problem 8

Problems(6)