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In phoenix, a Galaxy far, far away, there are $2021$ planets. Define a $fire$ to be a path between two objects in phoenix. It is known that between every pair of planets either a single fire burns or no burning occurs. If we consider any subset of $2019$ planets, the total number of fires burning be...

Asif Hossain wrote: ↑

Thu Apr 01, 2021 12:15 pm

Enthurelxyz wrote: ↑

Wed Mar 31, 2021 11:17 am

Let do a thing: you'll try this problem for 20 minutes. If you can solve the problem then give the solution. If you can't, you can write how you have approached the problem for 20 minutes

off topic :
hmmm i see you are very competitve

Is it bad?

Caitlin is playing an game on a $3*3$ board. She starts her marker in the upper-lest square, and each turn she randomly chooses an adjacent square to move her marker to.(She can’t move diagonally.) If she moves to the center square, she fights the dragon. After Caitlin has moved her marker four time...

User Mehrab4226 has already given the solution, though here I have shown a similar approach. Firstly, let's look at how many sets there are with $5$ elements that satisfy the question. It is easy to see there are five sets, namely $\{1, 2, 3, 4, 5\}, \{2, 3, 4, 5, 6\}, \{3, 4, 5, 6, 7\}, \{4, 5, 6,...

Let us denote that kind of subsets as $X$ At first, we look at how many sets of $5$ consecutive numbers are there. 1.$\{1,2,3,4,5\}$ 2.$\{2,3,4,5,6\}$ 3.$\{3,4,5,6,7\}$ 4.$\{4,5,6,7,8\}$ 5.$\{5,6,7,8,9\}$ Now we will divide our work in $5$ cases. Case $1$ represents when number$ 1$. of the list is ...

How many subsets of {1,2,3,4,5,6,7,8,9} contain $5$ consecutive numbers? Let us denote that kind of subsets as $X$ At first, we look at how many sets of $5$ consecutive numbers are there. 1.$\{1,2,3,4,5\}$ 2.$\{2,3,4,5,6\}$ 3.$\{3,4,5,6,7\}$ 4.$\{4,5,6,7,8\}$ 5.$\{5,6,7,8,9\}$ Now we will divide ou...

How many subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ contain $5$ consecutive numbers?

Let $A$ be one of the common points of two intersecting circles. Through $A$ construct a line on which the two circles cut out equal chords.

Let, $n\neq 7$ So, $n \equiv 1,2,3,4,5,6 (mod$ $7)$ $ :arrow: $n^2 \equiv 1,2,4 (mod$ $7)$. If $n^2 \equiv 1 (mod$ $7)$ then $n ^2-8\equiv 0 (mod$ $7)$. So, $n^2=115$ but $15$ is not a perfect square. If $n^2\equiv 2 (mod$ $7)$ then $n^2-2=7 :arrow: n=3$ $but $3^2-8=1$ which is not a prime. If $n^2 ...

Draw two lines $l_1$ and $l_2$ on P such that they are perpendicular to $AD,BC$ and $AB,CD$ respectively. $l_1$ intersects $AD,BC$ at $E,F$ respectively and $l_2$ intersects $AB,CD$ at $G,H$ respectively. As, $\angle DAP = \angle DCP$ :arrow: $ \angle EAP=\angle HCP$ :arrow: $ \triangle AEP$ is simi...