## Search found 17 matches

Tue Feb 09, 2021 8:02 pm
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 ...
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\$.
Wed Jan 20, 2021 10:54 am
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...
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.
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...
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?
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।