Solve for \(x\):

\[ \frac{ x} { x- 1 } = \frac{1}{x-1}. \]

If you place a quarter at room temperature in a block of dry ice (as above), the quarter will vibrate as though it is shivering. Why does this happen?

\(\)

A public auditorium is open from 6 AM to 9 PM for organizations to host their events. Each organization submits start and finish times for their event, which are displayed below:

Since events cannot be hosted concurrently, not all of the proposals can be accepted. To host the largest number of events, which of the following selection criteria should be used?

**Note:**

- The rows are merely used to organize the information in a clear manner. It does not imply that the entire row has to be selected.
- If two events share a common point on the time axis, then both proposals cannot be accepted at the same time.
- In the case that two or more events have the same number of overlap, finish time, or duration, then ties are broken arbitrarily.

The calculator above takes in an integer input at the lower left. Then, a path is taken to the top right - at each step moving up or right - and the operations given are performed on the input sequentially.

If a path gives an output of 43, which of the following choices could have been the initial input?

Is it possible to place distinct digits (0 to 9) in the boxes so that the 4 equations hold true?

Note: Since there are only 8 boxes, two digits will be left over.

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