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2017-04-31 Advanced

         

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.

\[\begin{cases} a^{ 2 }+b^{ 2 }+c^{ 2 }+d^{ 2 }=1 \\\\ K=ab+ac+ad+bc+bd+3cd \end{cases} \]

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.

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:

  • In a \(k\)-regular graph, each vertex has \(k\) neighbors.
  • An \(n\)-hypercube graph is the graph of an \(n\)-dimensional hypercube. In other words, its vertices are elements of \(\{0,1\}^n\) (that is, \(n\)-tuples with each component 0 or 1), and two vertices are adjacent if and only if they differ in exactly one component.
  • The labeling is not necessarily a proper vertex coloring: two adjacent vertices may have the same label.

\[ a^n-b^{n+2}=(a+b)^{n-1}\]

\(a\) 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 \(p^3-q^5=(p+q)^2.\)

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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