## Beginner's Marathon

For students of class 6-8 (age 12 to 14)
aritra barua
Posts: 57
Joined: Sun Dec 11, 2016 2:01 pm

### Re: Beginner's Marathon

Could you please assert the last line more clearly?

Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

### Re: Beginner's Marathon

The numbers of irreducible fractions is even. For example: if $n=6$,

$\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6}$. Here, $\frac{1}{6}$ and $\frac{5}{6}$ is irreducible and there number is tow, in fact even.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

### Re: Beginner's Marathon

aritra barua wrote:Let us denote by $ABC$,the area of $\bigtriangleup ABC$
Use $[ABC]$ to denote the area of $\triangle ABC$. It is more natural and used frequently.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

aritra barua
Posts: 57
Joined: Sun Dec 11, 2016 2:01 pm

### Re: Beginner's Marathon

The proof is trivial for some $n$ which has the form $2k$ where $k$ is an integer.Now,to say,this problem is equivalent to proving that the number of reducible fractions is even when $n$ is odd because the aforementioned sequence has precisely 'n-1' fractions(I.g:even numbered fractions).Now,we take $n$ as composite because there is nothing to prove when $n$ is a prime.Let us take a reducible fraction $\frac{n-i}{n}$ when g is a common divisor of 'n' and 'n-i'.This means that g|I and thus there is another fraction which is reducible $\frac{i}{n}$ exclusive of $\frac{g}{n}$ and $\frac{n-g}{n}$.But that follows that every time we choose a reducible fraction,there would be another reducible fraction and thus the number of reducible fractions will be even.It also says that the number of irreducible fractions will also be even.

Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

### Re: Beginner's Marathon

$\text{Problem 16}$

There are $n$ ants on a stick of length one unit, each facing left or right. At time $t = 0$, each ant starts moving with a speed of $1$ unit per second in the direction it is facing. If an ant reaches the end of the stick, it falls off and doesn’t reappear. When two ants moving in opposite directions collide, they both turn around and continue moving with the same speed (but in the opposite direction). Show that all ants will fall off the stick in at most $1$ second.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

Epshita32
Posts: 37
Joined: Mon Aug 24, 2015 12:34 am

### Re: Beginner's Marathon

As the ants are arbitrary, when they collide and move in opposite directions, basically nothing changes. For example, let's name them A and B. A was going right and B left. When they collide, they change directions, so A goes left and B goes right. But as the ants are similar, we can just take it, that they are continuing in the orignal path.
So, now, the longest path an ant can cross is if it is on one end of a stick and going in the opposite direction. Then, the stick is 1 unit long and the speed of the ant is 1 unit per second.
So, the max time the ants will take to fall off the stick is 1 second.

aritra barua
Posts: 57
Joined: Sun Dec 11, 2016 2:01 pm

### Re: Beginner's Marathon

Problem $17$ : Let $\bigtriangleup$ $ABC$ be an acute triangle.$D$ is the foot of perpendicular drawn from $C$ on $AB$.Let the bisector of $\angle$ $ABC$ intersect $CD$ at $E$ and $\bigcirc$ $ADE$ at $F$.If $\angle$ $ADF$ =45˚,prove that $CF$ is tangent to $\bigcirc$ $ADE$.

Atonu Roy Chowdhury
Posts: 64
Joined: Fri Aug 05, 2016 7:57 pm

### Re: Beginner's Marathon

aritra barua wrote:Problem $17$ : Let $\bigtriangleup$ $ABC$ be an acute triangle.$D$ is the foot of perpendicular drawn from $C$ on $AB$.Let the bisector of $\angle$ $ABC$ intersect $CD$ at $E$ and $\bigcirc$ $ADE$ at $F$.If $\angle$ $ADF$ =45˚,prove that $CF$ is tangent to $\bigcirc$ $ADE$.
My solution:
This was freedom. Losing all hope was freedom.

Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

### Re: Beginner's Marathon

I think you should give the full solution to the problems or give hints if the thread is stuck for some days.
Frankly, my dear, I don't give a damn.

Problem $18$
You and your spouse invited $10$ couples to a party. In that party, some of them shook hands with others. But no one shook hands with their spouse. After the party, you asked everyone(including your wife) how many person they shook hands with. You found that everyone shook hands with a different number of people. How many people did your wide shake hand with?