Found problems: 5
2018 Iranian Geometry Olympiad, 1
As shown below, there is a $40\times30$ paper with a filled $10\times5$ 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.
[i]Proposed by Morteza Saghafian[/i]
2018 Iranian Geometry Olympiad, 2
Convex hexagon $A_1A_2A_3A_4A_5A_6$ lies in the interior of convex hexagon $B_1B_2B_3B_4B_5B_6$ such that $A_1A_2 \parallel B_1B_2$, $A_2A_3 \parallel B_2B_3$,..., $A_6A_1 \parallel B_6B_1$. Prove that the areas of simple hexagons $A_1B_2A_3B_4A_5B_6$ and $B_1A_2B_3A_4B_5A_6$ are equal. (A simple hexagon is a hexagon which does not intersect itself.)
[i]Proposed by Hirad Aalipanah - Mahdi Etesamifard[/i]
2018 Iranian Geometry Olympiad, 4
There are two circles with centers $O_1,O_2$ lie inside of circle $\omega$ and are tangent to it. Chord $AB$ of $\omega$ is tangent to these two circles such that they lie on opposite sides of this chord. Prove that $\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ$.
[i]Proposed by Iman Maghsoudi[/i]
2018 Iranian Geometry Olympiad, 3
In the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$.
[i]Proposed by Mahdi Etesamifard[/i]
2018 Iranian Geometry Olympiad, 5
There are some segments on the plane such that no two of them intersect each other (even at the ending points). We say segment $AB$ [b]breaks[/b] segment $CD$ if the extension of $AB$ cuts $CD$ at some point between $C$ and $D$.
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -5.267474904743955, xmax = 11.572179069738377, ymin = -10.642621257034536, ymax = 4.543526642434019; /* image dimensions */
/* draw figures */
draw((-4,-2)--(1.08,-2.03), linewidth(2));
draw(shift((-2.1866176795507295,-2.0107089507113147))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((-0.16981767035094117,3.225314210196242)--(-2.1866176795507295,-2.0107089507113147), linewidth(2) + linetype("4 4"));
draw((-0.16981767035094117,3.225314210196242)--(-0.8194002739586808,1.538865607509914), linewidth(2));
label("$A$",(-1.2684397405642523,3.860690076971137),SE*labelscalefactor,fontsize(16));
label("$B$",(-1.9211395070170559,2.002590777612728),SE*labelscalefactor,fontsize(16));
label("$C$",(-4.971261820527631,-1.6571211388676117),SE*labelscalefactor,fontsize(16));
label("$D$",(1.08925640451367566,-1.6571211388676117),SE*labelscalefactor,fontsize(16));
/* dots and labels */
dot((-4,-2),dotstyle);
dot((1.08,-2.03),dotstyle);
dot((-0.16981767035094117,3.225314210196242),dotstyle);
dot((-0.8194002739586808,1.538865607509914),dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
$a)$ Is it possible that each segment when extended from both ends, breaks exactly one other segment from each way?
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -6.8, xmax = 8.68, ymin = -10.32, ymax = 3.64; /* image dimensions */
/* draw figures */
draw((-2.56,1.24)--(-0.36,1.4), linewidth(2));
draw((-3.32,-2.68)--(-1.24,-3.08), linewidth(2));
draw(shift((-2.551651190956802,-2.8277593863544612))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw(shift((-0.8889576602618603,1.3615303519809556))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((-2.551651190956802,-2.8277593863544612)--(-0.8889576602618603,1.3615303519809556), linewidth(2) + linetype("4 4"));
draw((-1.4097008194020806,0.049476186483185636)--(-1.8514772275312024,-1.0636149148218605), linewidth(2));
/* dots and labels */
dot((-2.56,1.24),dotstyle);
dot((-0.36,1.4),dotstyle);
dot((-3.32,-2.68),dotstyle);
dot((-1.24,-3.08),dotstyle);
dot((-1.4097008194020806,0.049476186483185636),dotstyle);
dot((-1.8514772275312024,-1.0636149148218605),dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
$b)$ A segment is called [b]surrounded[/b] if from both sides of it, there is exactly one segment that breaks it.\\
([i]e.g.[/i] segment $AB$ in the figure.) Is it possible to have all segments to be surrounded?
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(7cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -10.70976151557872, xmax = 18.64292748469251, ymin = -16.354300717041443, ymax = 9.136192362141452; /* image dimensions */
/* draw figures */
draw((1.0313140845297686,0.748205038977829)--(-1.3,-4), linewidth(2.8));
draw((-5.780195085389632,-2.13088646583346)--(-2.549994860479401,-2.13088646583346), linewidth(2.8));
draw((4.121070821400425,-3.816208322308361)--(1.78,-1.88), linewidth(2.8));
draw(shift((-0.38228674372374466,-2.13088646583346))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((-2.549994860479401,-2.13088646583346)--(-0.38228674372374466,-2.13088646583346), linewidth(2.8) + linetype("4 4"));
draw(shift((0.32979226045261084,-0.6805897691262632))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((4.121070821400425,-3.816208322308361)--(0.32979226045261084,-0.6805897691262632), linewidth(2.8) + linetype("4 4"));
draw((-3.6313140845297687,-8.74820503897783)--(3.600422205681574,5.980726991931396), linewidth(2.8) + linetype("2 2"));
label("$A$",(-0.397698406272906,1.754593418658662),SE*labelscalefactor,fontsize(16));
label("$B$",(-2.6377720405041316,-3.266261278756151),SE*labelscalefactor,fontsize(16));
/* dots and labels */
dot((1.0313140845297686,0.748205038977829),linewidth(6pt) + dotstyle);
dot((-1.3,-4),linewidth(6pt) + dotstyle);
dot((-5.780195085389632,-2.13088646583346),linewidth(6pt) + dotstyle);
dot((-2.549994860479401,-2.13088646583346),linewidth(6pt) + dotstyle);
dot((4.121070821400425,-3.816208322308361),linewidth(6pt) + dotstyle);
dot((1.78,-1.88),linewidth(6pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
[i]Proposed by Morteza Saghafian[/i]