MathDB

3

Part of 2013 Saudi Arabia GMO TST

Problems(3)

there are exactly 29 non-similar regular n-pointed stars

Source: 2013 Saudi Arabia GMO TST I p3

7/26/2020
Define a regular nn-pointed star to be a union of nn lines segments P1P2,P2P3,...,PnP1P_1P_2, P_2P_3, ..., P_nP_1 such that ∙\bullet the points P1,P2,...,PnP_1,P_2,...,P_n are coplanar and no three of them are collinear, ∙\bullet each of the nn line segments intersects at least one of the other line segments at a point other than an endpoint, ∙\bullet all of the angles at P1,P2,...,PnP_1, P_2,..., P_n are congruent , ∙\bullet all of the nn line segments P1P2,P2P3,...,PnP1P_1P_2, P_2P_3, ..., P_nP_1 are congruent, and ∙\bullet the path P1P2...PnP1P_1P_2...P_nP_1 turns counterclockwise at an angle less than 180o180^o at each vertex. There are no regular 33-pointed, 44-pointed, or 66-pointed stars. All regular 55-pointed star are similar, but there are two non-similar regular 77-pointed stars. Find all possible values of nn such that there are exactly 2929 non-similar regular nn-pointed stars.
combinatoricscombinatorial geometry
largest integer k such that k divides n^{55} - n

Source: 2013 Saudi Arabia GMO TST II p3

7/26/2020
Find the largest integer kk such that kk divides n55−nn^{55} - n for all integer nn.
dividesdivisiblenumber theory
angle chasing given AB = ID and AH = OH , orthocenter, incenter, circumcenter

Source: 2013 Saudi Arabia GMO TST day III p3

7/31/2020
ABCABC is a triangle, HH its orthocenter, II its incenter, OO its circumcenter and ω\omega its circumcircle. Line CICI intersects circle ω\omega at point DD different from CC. Assume that AB=IDAB = ID and AH=OHAH = OH. Find the angles of triangle ABCABC.
geometryincenterCircumcenterorthocenterequal segmentscircumcircle