3 people have each been assigned \(H\) or \(T\) on their foreheads, based on the results of tossing a fair coin. Each member can see each others' letters but not their own. Their common goal is to win a game with the following rules:

- They must all make a statement at the same time, and no communication is allowed beforehand.
- Each member's statement can either be a guess of their own letter ("H" or "T") or "pass".
- They win if at least one person guesses correctly and no one guesses incorrectly.

With everyone applying the optimal strategy, what is their probability of winning?

Let \(f : \mathbb{Z}^+ \longrightarrow \mathbb{Z}^*\) the function that assigns to each positive integer \(n\) the number of rational numbers \(\frac{p}{q}\) such that \[\begin{equation} \left\lbrace \begin{array}{ccc} p+q = n \\ 0<\frac{p}{q}<1 \\ \gcd(p,q)=1. \end{array}\right. \end{equation}\] For example, when \(n=8,\) we have \(2\) such rational numbers: \(\frac{1}{7}\) and \(\frac{3}{5}\). Hence \(f(8)=2\).

What is the first positive integer \(m\) such that there is no solution to \(f(n)=m?\)

As shown in the diagram, using 2 straight cuts from the same starting point on the perimeter, I've cut a circular pizza into 3 pieces of equal areas. However, each of the 2 identical pieces at the top and bottom has more crust than the middle piece!

A slice with larger crust has \(X\)% more crust than the slice with smaller crust.

Assuming the crust of the pizza is 1-dimensional, find \(X\) to the nearest integer.

In a Cartesian coordinate system, there are 2 spheres and 1 plane described as follows:

- \(S_1:\) Centered at the origin with radius 1.
- \(S_2:\) Centered at \((10,2,1)\) with radius 3.
- \(P:\) Defined by the equation \(9x-10z=260.\)

The first sphere \(S_1\) acts as a light source, the second sphere \(S_2\) is opaque, and the plane \(P\) is a screen.

What is the area of the umbra (the darkest part of the shadow, where an observer cannot see the light source at all) on the screen?

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