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 $\left\lbrace \begin{array}{ccc} p+q = n \\ 0<\frac{p}{q}<1 \\ \gcd(p,q)=1. \end{array}\right.$ 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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