MathDB

1

Part of 2018 Iranian Geometry Olympiad

Problems(3)

2018 IGO Elementary level P1

Source:

9/20/2018
As shown below, there is a 40×3040\times30 paper with a filled 10×510\times5 rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is to minimize the total length of the straight cuts. How to achieve this goal, and what is that minimized length? Show the correct cuts and write the final answer. There is no need to prove the answer.
Proposed by Morteza Saghafian
IGO2018 igoIrangeometry
fig with 3 rectangles & 2 perpendiculars on large diagonal given, computational

Source: Iranian Geometry Olympiad 2018 IGO Intermediate p1

9/19/2018
There are three rectangles in the following figure. The lengths of some segments are shown. Find the length of the segment XYXY . https://2.bp.blogspot.com/-x7GQfMFHzAQ/W6K57utTEkI/AAAAAAAAJFQ/1-5WhhuerMEJwDnWB09sTemNLdJX7_OOQCK4BGAYYCw/s320/igo%2B2018%2Bintermediate%2Bp1.png Proposed by Hirad Aalipanah
geometryrectangleComputational
equal segments starting with 2 intersecting circles

Source: Iranian Geometry Olympiad 2018 IGO Advanced p1

9/19/2018
Two circles ω1,ω2\omega_1,\omega_2 intersect each other at points A,BA,B. Let PQPQ be a common tangent line of these two circles with P∈ω1P \in \omega_1 and Q∈ω2Q \in \omega_2. An arbitrary point XX lies on ω1\omega_1. Line AXAX intersects ω2 \omega_2 for the second time at YY . Point Y′≠YY'\ne Y lies on ω2\omega_2 such that QY=QY′QY = QY'. Line Y′BY'B intersects ω1 \omega_1 for the second time at X′X'. Prove that PX=PX′PX = PX'.
Proposed by Morteza Saghafian
geometrycirclesequal segments