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## Search found 281 matches

A probelm is posted twice. So, actually this is the number 50. Problem 50: Let the incircle touches side $BC$ of a triangle $\triangle ABC$ at point $D$. Let $H$ be the orthocenter of $\triangle ABC$ and $M$ be the midpoint of segment $AH$. Let $E$ be a point on $AD$ so that $HE \perp AD$. Let $ME \... Fri Dec 01, 2017 9:29 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 108 Views: 4745 ### Re: Geometry Marathon : Season 3 Problem 47: Let$ABCD$be a cyclic quadrilateral.$AB$intersects$DC$at$E$.$AD$intersects$BC$at$F$. Let$M, N, P$are midpoints of$BD, AC, EF$respectively. Prove that$PN.PM=PE^2$Mon Nov 20, 2017 12:32 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 108 Views: 4745 ### Re: Geometry Marathon : Season 3$\text{Problem 45}$Let$ABC$be a triangle with orthocentre$H$and circumcircle$\omega$centered at$O$. Let$M_a,M_b,M_c$be the midpoints of$BC,CA,AB$. Lines$AM_a,BM_b,CM_c$meet$\omega$again at$P_a,P_b,P_c$. Rays$M_aH,M_bH,M_cH$intersect$\omega$at$Q_a,Q_b,Q_c$. Prove that$P_aQ_a,P_...
Tue Oct 31, 2017 12:01 am

Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 108
Views: 4745

For the sake of the contradiction, let's assume that it does. Then, $a_1 \equiv a_1. a_2 \equiv a_1. a_2. a_3$ $\equiv........\equiv a_1. a_2......a_{k-2}. a_k$ $\equiv.....\equiv a_1. a_k \equiv a_k \pmod n$. So, $n$ divides $|a_1 - a_k|$. But $0 < |a_1 - a_k| < n$, which is a contradiction. So, $... Tue Sep 05, 2017 12:25 am Forum: Algebra Topic: Sequence and divisibility Replies: 2 Views: 295 ### Re: The Gonit IshChool Project - Beta Name you'd like to be called: Tanmoy Course you want to learn: Functional Equations and Number Theory Problem solving. Preferred methods of communication (Forum, Messenger, Telegram, etc.):Telegram. Do you want to take lessons through PMs or Public?: Public Wed Mar 29, 2017 7:28 pm Forum: National Math Camp Topic: The Gonit IshChool Project - Beta Replies: 15 Views: 789 ### Re: Combi Marathon$\text {Problem 8}$We have$\dfrac {n(n+1)} {2}$stones in$k$piles. In each move we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, after that stone is removed the pile vanishes). Show that regardless of the initial configuration, we always end u... Fri Feb 24, 2017 9:21 pm Forum: Combinatorics Topic: Combi Marathon Replies: 43 Views: 2556 ### Re: IMO Marathon$\text{Problem 55}$Let$n$be a positive integer and let$(a_1,a_2,\ldots ,a_{2n})$be a permutation of$1,2,\ldots ,2n$such that the numbers$|a_{i+1}-a_i|$are pairwise distinct for$i=1,\ldots ,2n-1$. Prove that$\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$if and only if$a_1-a_{2n}=n$. Fri Feb 24, 2017 5:09 pm Forum: International Mathematical Olympiad (IMO) Topic: IMO Marathon Replies: 184 Views: 23335 ### Re: Combi Marathon$\text{Problem 7}$Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set$S$of$n>1$collinear points. Then, for every unordered pair of points$\{X,Y\}$in$S$, Elmo draws the circle with diameter$XY$so that each pair of circles which intersect at two distinct points ar... Fri Feb 24, 2017 4:27 pm Forum: Combinatorics Topic: Combi Marathon Replies: 43 Views: 2556 ### Re: IMO Marathon$\text{Problem 54}$The following operation is allowed on a finite graph: choose any cycle of length$4$(if one exists), choose an arbitrary edge in that cycle, and delete this edge from the graph. For a fixed integer$n \ge 4$, find the least number of edges of a graph that can be obtained by rep... Fri Feb 24, 2017 1:25 pm Forum: International Mathematical Olympiad (IMO) Topic: IMO Marathon Replies: 184 Views: 23335 ### Re: Combi Solution Writing Threadie Problem 1 Let$n > 3$be a fixed positive integer. Given a set$S$of$n$points$P_1, P_2,\cdots, P_n$in the plane such that no three are collinear and no four concyclic, let$a_t$be the number of circles$P_i P_j P_k$that contain$P_t$in their interior, and let$m(S) = a_1 + a_2 +\cdots + a_n...
Mon Feb 20, 2017 12:05 am

Forum: Combinatorics