$\text{Problem 55}$

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 $a_1-a_{2n}=n$.

## Search found 305 matches

- Fri Feb 24, 2017 5:09 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**63670**

- 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
- Forum: International Mathematical Olympiad (IMO)
- 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
- Topic: Combi Solution Writing Threadie
- Replies:
**10** - Views:
**9416**

### 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...

- 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

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 $BP \perp CP$.

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**

### Re: Geometry Marathon : Season 3

$\text{Problem 3:}$ In Acute angled triangle $ABC$, let $D$ be the point where $A$ angle bisector meets $BC$. The perpendicular from $B$ to $AD$ meets the circumcircle of $ABD$ at $E$. If $O$ is the circumcentre of triangle $ABC$ then prove that $A,E$ and $O$ are collinear. $\text{Another Solution:...

- Fri Jan 06, 2017 11:58 am
- Forum: Secondary Level
- Topic: TJMO 1996/2
- Replies:
**4** - Views:
**2005**

### Re: TJMO 1996/2

Our counting system issiwomcre wrote:What is trinary?

**Decimal**. It has $10$ digits: $0,1,2,3,4,5,6,7,8,9$.

**Trinary**is another system of number counting. But it has only three digits: $0,1$ and $2$.