Search found 16 matches

by M Ahsan Al Mahir
Wed Jan 09, 2019 11:46 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2018 P3
Replies: 0
Views: 5475

IMO 2018 P3

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains ...
by M Ahsan Al Mahir
Wed Jan 09, 2019 11:46 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2018 P6
Replies: 0
Views: 5333

IMO 2018 P6

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\]Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$.
by M Ahsan Al Mahir
Wed Jan 09, 2019 11:45 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2018 P5
Replies: 0
Views: 5382

IMO 2018 P5

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$...
by M Ahsan Al Mahir
Wed Jan 09, 2019 11:45 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2018 P2
Replies: 0
Views: 5494

IMO 2018 P2

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.
by M Ahsan Al Mahir
Wed Jan 09, 2019 11:44 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2018 P4
Replies: 0
Views: 5298

IMO 2018 P4

A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the...
by M Ahsan Al Mahir
Wed Jan 09, 2019 11:44 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2018 P1
Replies: 1
Views: 5039

IMO 2018 P1

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ ...
by M Ahsan Al Mahir
Fri Mar 31, 2017 10:09 pm
Forum: Social Lounge
Topic: Math
Replies: 7
Views: 5194

Re: Math

you drank it??
by M Ahsan Al Mahir
Wed Mar 29, 2017 9:34 pm
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 18765

Re: Beginner's Marathon

Solution to P6: Let $ME$ meet $AC$ at $X$, $MF$ meet $AB$ at $Y$. Let the circumcircles of $\triangle CEX$, $\triangle CFY$ meet at $T$. $\angle FYC = \angle FTC = \angle ETC = \angle EXC = 90$. So $E, F, T$ are colliner. As $MX \cdot ME = MA^2 = MB^2 = MY \cdot MF$, $M$ lies on the radical axis of...
by M Ahsan Al Mahir
Tue Mar 28, 2017 4:57 pm
Forum: National Math Camp
Topic: The Gonit IshChool Project - Beta
Replies: 28
Views: 31235

Re: The Gonit IshChool Project - Beta

Name you'd like to be called: Lazim
Course you want to learn: Functional Equation, Number Theory
Preferred methods of communication (Forum, Messenger, Telegram, etc.): Telegram
Do you want to take lessons through PMs or Public?: Public
by M Ahsan Al Mahir
Mon Feb 27, 2017 1:00 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 54561

Re: Geometry Marathon : Season 3

$\text{Problem 36:}$ Let $ABC$ be a triangle and $O$ be its circumcenter. A point $P$ is on the internal angle bisector of $\angle B$. Let $(P)$ be the circle that touches $BC$ and $BA$ at $X, Y$. Prove that the reflection of $OP$ wrt $XY$ passes throught the midpoint of $BH$. I request Geodip bro ...