MathDB

shortest path in a grid

Source: IGO Elementary 2016 1

7/22/2018

Ali wants to move from point

A

to point

B

. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between

A

and

B

. Only draw the path and write its length. https://1.bp.blogspot.com/-nZrxJLfIAp8/W1RyCdnhl3I/AAAAAAAAIzQ/NM3t5EtJWMcWQS0ig0IghSo54DQUBH5hwCK4BGAYYCw/s1600/igo%2B2016.el1.png by Morteza Saghafian

geometryElementary

Secant less than half trapezoid perimeter

Source: Iranian Geometry Olympiad 2016 Medium 1

5/26/2017

In trapezoid

ABCD

with

AB || CD

,

\omega_1

and

\omega_2

are two circles with diameters

AD

and

BC

, respectively. Let

X

and

Y

be two arbitrary points on

\omega_1

and

\omega_2

, respectively. Show that the length of segment

XY

is not more than half the perimeter of

ABCD

.
Proposed by Mahdi Etesami Fard

geometrytrapezoidperimeter

3 points are collinear!

Source: IGO 2016,Advanced level,P1

9/13/2016

Let the circles

\omega

and

\omega^ \prime

intersect in

A

and

B

. Tangent to circle

\omega

at

A

intersects

\omega^ \prime

in

C

and tangent to circle

\omega^ \prime

at

A

intersects

\omega

in

D

. Suppose that

CD

intersects

\omega

and

\omega^ \prime

in

E

and

F

, respectively (assume that

E

is between

F

and

C

). The perpendicular to

AC

from

E

intersects

\omega^ \prime

in point

P

and perpendicular to

AD

from

F

intersects

\omega

in point

Q

(The points

A, P

and

Q

lie on the same side of the line

CD

). Prove that the points

A, P

and

Q

are collinear. Proposed by Mahdi Etesami Fard

geometrygeometry proposed

1

Problems(3)