BdMO National 2014: Junior 10

[Labib]

Oindri has $100$ chocolates. She finished eating all her chocolates in $58$ days by eating at least one chocolate each day. Prove that, in some consecutive days she eat exactly $15$ chocolates.

[Labib]

I'll give a step by step solution and I encourage everyone to think a lot before looking at the steps since it's an amazing problem.

Step $1$:

Most people actually do not understand what the problem is asking for (Even I did not get it at first).
Let's define $d_i$ as the number of chocolates the girl eats on the $i^{th}$ day. You have to prove that, $d_i + d_{i+1} + ... + d_j = 15$ for some $i$ and $j$, where $1 \leq i\leq j \leq 58$.

Step $2$:

Let's define $a_j$ in the following way,
\[a_j = \sum_{i=1}^{j}d_i\] where $1\leq j\leq 58$
Since $d_i \geq 1$ for all $1 \leq i \leq 58$,
$a_i < a_j$ for all $i < j$.
Again, we define $b_i = a_i + 15$ for all $1 \leq i \leq 58$.
Since, $a_i < a_j$ for all $i < j$, therefore, $b_i < b_j$ for all $i < j$.
Now all we have to prove is that, for some $i$ and $j$, $a_i = b_j$ (where $1\leq i,j \leq 58$).
Do you know why? Can you prove it?

Step $3$:

One important thing you may notice here is that this problem can now be solved by PHP(Pigeon hole principle).
The minimum possible value of $a_i$ for any $i$ is $1$ and the maximum possible value of $b_j$ for any $j$ is $100+15 = 115$.
So, $1 \leq a_i, b_i \leq 115$ holds for all $1 \leq i \leq 58$.
But the sequences $a_i$ and $b_i$ both have $58$ different terms. So there are, in total, $58+58=116$ terms in the two sequences. But only $115$ distinct possible values for them, $2$ of the terms must be equal. Since $a_i \neq a_j$ and $b_i \neq b_j$ for all $i\neq j$, the two equal terms must be from the two different sequences.
So, there is some $i$ and $j$ such that $a_i = b_j \Rightarrow a_i-b_j=0\Rightarrow a_i-a_j=15$.
Thus, she ate $15$ chocolates in some consecutive days. [proved]

[atiab jobayer]

Labib, write in bangla. I'm not expert in English

[atiab jobayer]

Subrata has invented a new type of clock,according to which there are 15 hours in each day and 80 minutes in each hour.for example Subrata's clock shows 10.00 when the actual time is 16.00 in a traditional clock. If the time is 20.36 in a traditional clock,then what will be the time in Subrata's clock ?

[Labib]

Atiab, I've hardly used some English here. All the important things are written as equations.
Try focusing on the maths instead of getting bothered with the English. I believe you'll be just fine.
Still, if you have difficulty, let me know which step you didn't get.
And, please post new problems in different threads with proper name (specially, these BdMO problems)
Here's a new thread for Junior $03$.