## Search found 289 matches

Thu Oct 03, 2019 11:44 pm
Forum: Geometry
Topic: IGO official website
Replies: 2
Views: 985

### Re: IGO official website

Mubin Hasan wrote:
Mon Sep 16, 2019 5:53 am
Do you happen to know how can i get the resources and the past papers in english???
https://igo-official.ir
Thu Jul 18, 2019 11:13 pm
Topic: IMO 2019/P6
Replies: 0
Views: 353

### IMO 2019/P6

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of ...
Thu Jul 18, 2019 11:11 pm
Topic: IMO 2019/P5
Replies: 0
Views: 386

### IMO 2019/P5

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, ...
Thu Jul 18, 2019 11:05 pm
Topic: IMO 2019/P4
Replies: 1
Views: 479

### IMO 2019/P4

Find all pairs $(k,n)$ of positive integers such that $k!=(2^n-1)(2^n-2)\cdots(2^n-2^{n-1})$

Proposed by Gabriel Chicas Reyes, El Salvador
Thu Jul 18, 2019 11:02 pm
Topic: IMO 2019/P3
Replies: 5
Views: 427

### IMO 2019/P3

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but ...
Thu Jul 18, 2019 10:59 pm
Topic: IMO 2019/P2
Replies: 0
Views: 324

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\... Thu Jul 18, 2019 10:57 pm Forum: International Mathematical Olympiad (IMO) Topic: IMO 2019/P1 Replies: 1 Views: 363 ### IMO 2019/P1 Let$\mathbb{Z}$be the set of integers. Determine all functions$f: \mathbb{Z} \rightarrow \mathbb{Z}$such that, for all integers$a$and$b$,$f(2a)+2f(b)=f(f(a+b)).$Proposed by Liam Baker, South Africa Wed Jan 30, 2019 12:10 pm Forum: Asian Pacific Math Olympiad (APMO) Topic: APMO 2011 Problem 1 Replies: 1 Views: 443 ### Re: APMO 2011 Problem 1 Topic locked. Already Posted here: viewtopic.php?f=15&t=1003&p=4148&hilit= ... %2F1#p4148 Please use search function before posting anything Fri Dec 01, 2017 9:29 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 15047 ### Re: Geometry Marathon : Season 3 A probelm is posted twice. So, actually this is the number 50. Problem 50: Let the incircle touches side$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 \...
Mon Nov 20, 2017 12:32 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 15047

### 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 $PN.PM=PE^2$