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A rectangle is divided by four lines, as shown, with 4 of the resulting triangles having known areas. Find the area of the quadrilateral that is shaded in pink.
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$\begin{cases} a^{ 2 }+b^{ 2 }+c^{ 2 }+d^{ 2 }=1 \\\\ K=ab+ac+ad+bc+bd+3cd \end{cases}$
Let 4 real numbers $a$, $b$, $c$, and $d$ satisfy the two equations above. If the minimum and maximum values of $K$ are $m$ and $M,$ respectively, what is the value of $m+M ?$
Give your answer to 3 decimal places.
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A $k$-regular graph satisfies the neighborhood diversity condition if it is possible to label each vertex with one of $1, 2, 3, \ldots, k$ such that for each vertex, all its neighbors have different labels.
Determine the sum of all $n$ satisfying $1 \le n \le 1000$ such that the $n$-hypercube graph satisfies the neighborhood diversity condition.
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Clarification:
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$a^n-b^{n+2}=(a+b)^{n-1}$
$a$ and $b$ are relatively prime positive integers and $n\, (>1)$ is an integer. Find all solutions to the equation above and enter your answer as $\sum (a+b+n).$
This is a generalization of the problem from 1997 Russian Olympiad:
For prime numbers $p$ and $q,$ solve $p^3-q^5=(p+q)^2.$
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On a projective plane, there are points and lines. Points may or may not lie on lines. Given any two points on the plane, there exists a unique line passing through them. Any two lines intersect at exactly one point; that is, for any two lines on the plane, there is a unique point that lie on both of them.
Suppose $P$ is a set of points and $L$ is a set of lines on a projective plane. A pair $(p, l)$ with $p \in P, l \in L$ is called an incidence if $p$ lies on $l$. Let $I(n)$ be the maximum number of incidences among all $P, L$ such that $|P| = |L| = n$.
For example, $I(4) = 9$, achieved by having $P = \{p_1, p_2, p_3, p_4\}$, $L = \{l_1, l_2, l_3, l_4\}$, and having the set of incidences
$\big\{(p_2, l_1), (p_3, l_1), (p_4, l_1), (p_1, l_2), (p_2, l_2), (p_1, l_3), (p_3, l_3), (p_1, l_4), (p_4, l_4)\big\}.$
It can be proven that no configuration can reach more than 9 incidences. Geometrically, the above example can be represented in the following diagram:
However, remember that despite the objects being called points and lines, there is no requirement that they can actually be represented on the standard Euclidean plane.
As it turns out, $I(n) = \Theta(n^c)$ for some exponent $c$; in other words, there exist constants $c, N, C_1, C_2 > 0$ such that for all $n > N$, we have $C_1 \cdot n^c \le I(n) \le C_2 \cdot n^c$.
What is the value of $\lfloor 10000c \rfloor?$
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Notation: $\lfloor \cdot \rfloor$ denotes the floor function.
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