MathDB

equal segments , incenter and circumcircle related

Source: 1998 Estonia National Olympiad Final Round grade 9

3/11/2020

Let

S

be the incenter of the triangle

ABC

and let the line

AS

intersect the circumcircle of triangle

ABC

at point

D

(

D\ne A

). Prove that the segments

BD, CD

and

SD

are of equal length.

geometryincentercircumcircleequal segments

collinear wanted, semicircle related

Source: 1998 Estonia National Olympiad Final Round grade 10 p2

3/11/2020

Let

C

and

D

be two distinct points on a semicircle of diameter

AB

. Let

E

be the intersection of

AC

and

BD

,

F

be the intersection of

AD

and

BC

and

X, Y

, and

Z

are the midpoints of

AB, CD

, and

EF

, respectively. Prove that the points

X, Y,

and

Z

are collinear.

collineargeometrysemicircle

concurrency related to midpoints and incenters

Source: 1998 Estonia National Olympiad Final Round grade 11 p2

3/11/2020

In a triangle

ABC, A_1,B_1,C_1

are the midpoints of segments

BC,CA,AB, A_2,B_2,C_2

are the midpoints of segments

B_1C_1,C_1A_1,A_1B_1

, and

A_3,B_3,C_3

are the incenters of triangles

B_1AC_1,C_1BA_1,A_1CB_1

, respectively. Show that the lines

A_2A_3,B_2B_3

and

C_2C_3

are concurrent.

geometryincenterconcurrentmidpoints

prime numbers of the form 10101...01

Source: 1998 Estonia National Olympiad Final Round grade 12 p2

3/11/2020

Find all prime numbers of the form

10101...01

.

primeprimesDigitsnumber theory

2

Problems(4)