## Search found 16 matches

Wed Jan 09, 2019 11:46 pm
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 ...
Wed Jan 09, 2019 11:46 pm
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 $\angle{BXA} + \angle{DXC} = 180^\circ$.
Wed Jan 09, 2019 11:45 pm
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 $M$...
Wed Jan 09, 2019 11:45 pm
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 $i = 1, 2, \dots, n$.
Wed Jan 09, 2019 11:44 pm
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...
Wed Jan 09, 2019 11:44 pm
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 $FG$ ...
Fri Mar 31, 2017 10:09 pm
Forum: Social Lounge
Topic: Math
Replies: 7
Views: 5194

### Re: Math

you drank it??
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...
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
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 ...