Search found 17 matches

by Enthurelxyz
Tue Feb 09, 2021 8:02 pm
Forum: National Math Olympiad (BdMO)
Topic: BdMO National Junior 2020 P8
Replies: 1
Views: 312

Re: BdMO National Junior 2020 P8

As, $O$ is the center of $\triangle AFE$ and $\angle FAE=70$ so $\angle FOE=140$ so $ \angle OEF=20$. As, $AFDC$ is cyclic then $\angle FDB=70$. As, $AEDB$ is cyclic, it is easy to prove that $\angle EDC=70$. So, $\angle FDE=40$. $\angle FDE+\angle FOE=180$ so $OFDE$ is cyclic so $\angle OEF=\angle ...
by Enthurelxyz
Sun Jan 24, 2021 7:56 pm
Forum: Geometry
Topic: Reflections meet on circumcircle of ABC
Replies: 3
Views: 334

Reflections meet on circumcircle of ABC

Let $l$ is the Euler line of $\triangle ABC$. $l_a,l_b,l_c$ are the reflections of $l$ from $BC,AC,AB$ respectively. Prove that $l_a,l_b,l_c$ meet on the circumcircle of $ABC$.
by Enthurelxyz
Wed Jan 20, 2021 10:54 am
Forum: National Math Olympiad (BdMO)
Topic: BDMO National Junior 2018/6
Replies: 4
Views: 1473

Re: BDMO National Junior 2018/6

Let the $8$ lines are $l_1,l_2,\cdots ,l_8$. Take a point $O$ on the same plane. Now draw $8$ lines $L_i$ on point $O%$ such that $L_i \parallel l_i$. Now, on point $O$, there are $8$ different angles created by $L_1,L_2,\cdots ,L_8$. So, there should be at least an angle less than $23$. Let $\angle...
by Enthurelxyz
Mon Jan 18, 2021 12:31 pm
Forum: Combinatorics
Topic: Number of parallelograms in an equilateral triangle
Replies: 1
Views: 407

Number of parallelograms in an equilateral triangle

A triangular grid is obtained by tiling an equilateral triangle of side length $n$ by $n^2$ equilateral triangles of side length $1$. Determine the number of parallelograms bounded by line segments of the grid.
by Enthurelxyz
Sun Jan 10, 2021 1:12 pm
Forum: Number Theory
Topic: All eight integers are the same modulo 2005
Replies: 1
Views: 550

All eight integers are the same modulo 2005

At each corner of a cube, an integer is written. A $legal$ $transition$ of the cube consists in picking any corner of the cube and adding the value written at that corner to the value written at some adjacent corner. Prove that there is a finite sequence of $legal$ $transitions$ of the given cube su...
by Enthurelxyz
Thu Dec 17, 2020 1:50 pm
Forum: Site Support
Topic: How can I insert images in this forum?
Replies: 1
Views: 328

How can I insert images in this forum?

I am new in this forum. Sometimes I need to insert an image for a solution. How can I do it?
by Enthurelxyz
Sun Dec 06, 2020 7:12 pm
Forum: Higher Secondary Level
Topic: একজন আছে যে সবাইকে চিনে।
Replies: 3
Views: 1194

একজন আছে যে সবাইকে চিনে।

একটি কমিটিতে $2n+1$ সদস্য আছে। তাদের যে কোনো $n$ জনকে বাছাই করলে, বাকি $n+1$ জনের মধ্যে অন্তত একজন থাকবে যে এই $n$ জনের সবাইকে চিনে। প্রমাণ করো, কমিটিতে একজন আছে যিনি বাকি সবাইকে চিনে।