MathDB

1

Part of 2015 Iran Geometry Olympiad

Problems(3)

convex polygons from 4 wooden triangles with sides 3, 4, 5

Source: IGO 2015 Elementary 1

7/22/2018
We have four wooden triangles with sides 3,4,53, 4, 5 centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof)
A convex polygon is a polygon which all of it's angles are less than 180o180^o and there isn't any hole in it. For example: https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png
geometryconvexElementary
IGO-Intermediate-Problem 1

Source:

9/22/2015
Given a circle and Points P,B,AP,B,A on it.Point QQ is Interior of this circle such that: 1)1) ∠PAQ=90\angle PAQ=90. 2)PQ=BQ 2)PQ=BQ. Prove that ∠AQB−∠PQA=AB⌢\angle AQB - \angle PQA=\stackrel{\frown}{AB}.
proposed by Davoud Vakili, Iran.
geometry proposedgeometry
IGO - Hard level - problem 1

Source:

9/3/2015
let w1 w_1 and w2 w_2 two circles such that w1∩w2={A,B} w_1 \cap w_2 = \{ A , B \}
let X X a point on w2 w_2 and Y Y on w1 w_1 such that BY⊥BX BY \bot BX
suppose that O O is the center of w1 w_1 and X′=w2∩OX X' = w_2 \cap OX
now if K=w2∩X′Y K = w_2 \cap X'Y prove X X is the midpoint of arc AK AK
IGOgeometry