AMC 8, 2020, Problem Marathon

[Arisha Roy]

Problem 22

When a positive integer \(N\) is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of \(N=7\), the machine will output \(3 \cdot 7+1=22\). Then if the output is repeatedly inserted into the machine five more times, the final output is 26 .
\[
7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26
\]

When the same 6 -step process is applied to a different starting value of \(N\), the final output is 1 . What is the sum of all such integers \(N\) ?
\[
N \rightarrow \ldots \rightarrow \_\rightarrow-\rightarrow \_\rightarrow \rightarrow 1
\]

• This topic was modified 1 year, 4 months ago by Arisha Roy.
• This topic was modified 1 year, 4 months ago by Arisha Roy.
• This topic was modified 1 year, 4 months ago by Arisha Roy.
• This topic was modified 1 year, 4 months ago by Arisha Roy.

[Arisha Roy]

Solution:

Here the final output is 1 and we do the calculation in backward.

Here we will consider all possible inputs from which we can get the particular output.
\[
\{1\} \rightarrow\{2\} \rightarrow\{4\} \rightarrow\{1,8\} \rightarrow\{2,16\} \rightarrow\{4,5,32\} \rightarrow\{1,8,10,64\}
\]
Here 2 must come from 4 (as there is no integer \(n\) satisfying \(3 n+1=2\) )

But 16 could come from 32 or 5.

If we consider input is 5, output will be \(3 \ times 5 +1=16\).

If we consider input is 16, output will be \(\frac{32}{2}=16 \).

After 6 steps in the last set there are 4 numbers.

Sum of the numbers is \(1+8+10+64=83\).

• This reply was modified 1 year, 4 months ago by Arisha Roy.

[Arisha Roy]

Problem 23

Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?

[Arisha Roy]

Solution of problem 23

It is not possible for any student to receive 4 or 5 awards because this means that one of the other students receives no award. Thus, each student must receive either 1,2 , or 3 awards.

Case 1:

If a student receives 3 awards, then each of the other two students must receive 1 award.

If a student receives 3 awards, there are 3 ways to choose which student this is.

No of ways of choosing 3 awards among 5 awards=\(\left(\begin{array}{l}5 \\ 3\end{array}\right)\)

Now, there are 2 students and 2 awards left.

There are 2 ways to distribute 2 awards among 2 students.

For this case number of ways to distribute the awards= \(3 \cdot\left(\begin{array}{l}5 \\ 3\end{array}\right) \cdot 2=60\)

Case 2:

If a student receives 2 awards, then another student must receive 2 awards and the remaining student must receive 1 award.

We know there are 3 choices for the student who gets 1 award.

Then, there are \(\left(\begin{array}{l}5 \\ 2\end{array}\right)\) ways to give the first student his two awards and \(\left(\begin{array}{l}3 \\ 2\end{array}\right)\) ways to give the second student his 2 awards.

Finally, there is only 1 student and 1 award left, so there is only 1 way to distribute this award.

There are \(\left(\begin{array}{l}5 \\ 2\end{array}\right) \cdot\left(\begin{array}{l}3 \\ 2\end{array}\right) \cdot 1 \cdot 3=90\) ways to distribute the awards in this case.

Total number of ways will be \(60+90= 150 \).

[Arisha Roy]

Problem 25

Rectangles \(R_1\) and \(R_2\), and squares \(S_1, S_2\), and \(S_3\), shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of \(S_2\) in units?

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