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:
With everyone applying the optimal strategy, what is their probability of winning?
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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 \(f(n)=m?\)
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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.
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In a Cartesian coordinate system, there are 2 spheres and 1 plane described as follows:
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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Two parabolas (in blue) intersect at two points. The four tangent lines (in red) at their points of intersection form a \(3\times 4\) rectangle. Find the area of the closed region created by both parabolas.
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