2 circles can split the plane into at most 4 regions.

3 circles can split the plane into at most 8 regions.

4 circles can split the plane into at most \(\text{______}\) regions.

In a game of tennis, does having "deuce" (as compared to a simple "first to 4 points" system) help, hurt, or not affect the better player?

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**Details and Assumptions:**

- The "better player" wins each point with fixed probability \(p > 0.5.\)
- In "deuce" rules, if the game reaches 3-3, the winner is the first person to subsequently lead by 2 points (colloquially, "win by 2").

A \(20\times8\) rectangle is partitioned into three right triangles, and a circle is inscribed in each triangle.

Find the sum of the radii of these 3 circles.

There's a war on the playground, and you're the charismatic leader of the Greasers. The Socs, your sworn enemies, have fashioned themselves a defensive fort, behind which they're hiding their prized possession, the crown.

The only way to win the war is by making a direct hit on the crown with a tennis ball. Blocked from a direct hit by the fort, the Greasers have to shoot it off the wall behind the fort and bank it into the crown.

As shown in the diagram, the Greasers launch a tennis ball with speed \(v_1 = \SI[per-mode=symbol]{20}{\meter\per\second}\) at an angle of \(\theta=\SI{\frac{\pi}6}{\radian}\) with the horizontal. At the peak of its trajectory, the ball hits the wall and reflects with speed \(\frac12 v_1.\)

If the ball hits the crown as shown in the diagram, calculate the distance \(\ell\) (in \(\si{\meter}\)) between the launch point and the crown.

Assume that \(g = \SI[per-mode=symbol]{9.81}{\meter\per\second\squared}.\)

I have 7 distinct dollar bills, each of which is a distinct integer denomination of $50 or less. Using these bills, I can buy anything that costs an integer amount between $1 and $60 (inclusive) in exact change.

What is the least possible value of the largest denomination of these 7 bills (in dollars)?

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