## Search found 305 matches

Fri Feb 24, 2017 5:09 pm
Topic: IMO Marathon
Replies: 184
Views: 63670

### 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 4:27 pm
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27650

### 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 1:25 pm
Topic: IMO Marathon
Replies: 184
Views: 63670

### 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...
Mon Feb 20, 2017 12:05 am
Forum: Combinatorics
Replies: 10
Views: 9416

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 $m(S) = a_1 + a_2 +\cdots + a_n... Sun Feb 19, 2017 10:19 pm Forum: Combinatorics Topic: Combi Marathon Replies: 48 Views: 27650 ### Re: Combi Marathon Problem 1 Let$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...
Sat Jan 07, 2017 10:56 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63018

### Re: Geometry Marathon : Season 3

Problem 10:
Let $I$ be the incenter of $\triangle ABC$. The incircle touches $BC$ at $D$ and $K$ is the antipode of $D$ in $(I)$.
Let $M$ be the midpoint of $AI$. Prove that $KM$ passes through the Feuerbach Point.
Sat Jan 07, 2017 11:13 am
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63018

### Re: Geometry Marathon : Season 3

Problem 8: Given a cyclic quadrilateral $ABCD$ with circumcircle $(O)$. Let $AB \cap CD=E, \ AD \cap BC=F, \ AC \cap BD=G, \ AC \cap EF=P, \ BD \cap EF=Q$. Let $M, \ N$ be midpoints of $AC, \ BD$, respectively and let $MN \cap EF=H$. (i) Prove that $M, \ N, \ P, \ Q$ are concyclic. (ii) Let $K$ be ...
Fri Jan 06, 2017 10:09 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63018

### Re: Geometry Marathon : Season 3

Problem 5:
Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ (ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.
Fri Jan 06, 2017 9:46 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63018