Let $S = 1 + 11 + 111 + \cdots + 1\ldots 1$, where the last summand has $111$ decimal digits each equal to $1$. What is the sum of the digits of $S$?

Suppose that five particles travel back and forth on the unit interval $[0,1]$. At the start, all five particles move to the right with the same velocity. When a particle reaches $0$ or $1$, it reverses direction but maintains its speed. When two particles collide, they reverse direction and maintain speed. How many particle-particle collisions occur before the five particles occupy their original positions and are moving to the right?

In how many ways can a chess Queen move from one corner of the chess board to the opposite corner, moving closer to the goal square at every step?

What is the only positive integer $n$ such that there are exactly $n$ incongruent triangles with integer sides and perimeter $n$?

How many ways can you make one million dollars using any number of pennies, nickels, dimes, quarters, one-dollar bills, five-dollar bills, ten-dollar bills, twenty-dollar bills, fifty-dollar bills, and hundred-dollar bills?

What is noteworthy about the multisets $\{2,2\}$, $\{1,1,2,2,2,3,3,4\}$, and $\{1,1,1,2,2,3,3,3,4,6\}$? Can you characterize all multisets of positive integers with this property?

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