169 views

0 votes

Ten objects $\text{o1, o2,} \dots, \text{o10}$ were distributed among Amar, Barat, Charles, Disha, and Elise. Each item went to exactly one person. Each person got exactly two of the items, and this pair of objects is called her/his bundle.

The following table shows how each person values each object.

$$\begin{array} {|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & \textbf{o1} & \textbf{o2} & \textbf{o3} & \textbf{o4} & \textbf{o5} & \textbf{o6} & \textbf{o7} & \textbf{o8} & \textbf{o9} & \textbf{o10} \\\hline \textbf{Amar} & 4 & 9 & 9 & 3 & 7 & 3 & 8 & 7 & 9 & 5 \\\hline \textbf{Barat} & 5 & 9 & 7 & 5 & 5 & 3 & 6 & 8 & 10 & 8 \\\hline \textbf{Charles} & 8 & 8 & 8 & 3 & 6 & 4 & 5 & 8 & 9 & 6 \\\hline \textbf{Disha} & 8 & 8 & 8 & 5 & 5 & 3 & 6 & 4 & 9 & 8 \\\hline \textbf{Elise} & 6 & 8 & 9 & 5 & 6 & 5 & 6 & 3 & 7 & 10 \\\hline \end{array}$$

The value of any bundle by a person is the sum of that person’s values of the objects in that bundle. A person $\text{X}$ envies another person $\text{Y}$ if $\text{X}$ values $\text{Y’s}$ bundle more than $\text{X’s}$ own bundle.

For example, hypothetically suppose Amar’s bundle consists of $\text{o1}$ and $\text{o2},$ and Barat’s bundle consists of $\text{o3}$ and $\text{o4}.$ Then Amar values his own bundle at $4+9 = 13$ and Barat’s bundle at $9+3=12.$ Hence Amar does not envy Barat. On the other hand, Barat values his own bundle at $7+5=12$ and Amar’s bundle at $5+9=14.$ Hence Barat envies Amar.

The following facts are known about the actual distribution of the objects among the five people.

- If someone’s value for an object is $10,$ then she/he received that object.
- Objects $\text{o1, o2,}$ and $\text{o3}$ were given to three different people.
- Objects $\text{o1}$ and $\text{o8}$ were given to different people.
- Three people value their own bundles at $16.$ No one values her/his own bundle at a number higher than $16.$
- Disha values her own bundle at an odd number. All others value their own bundles at an even number.
- Some people who value their own bundles less than $16$ envy some other people who value their own bundle at $16.$ No one else envies others.

What is Amar’s value for his own bundle?