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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.

{a2+b2+c2+d2=1K=ab+ac+ad+bc+bd+3cd\begin{cases} a^{ 2 }+b^{ 2 }+c^{ 2 }+d^{ 2 }=1 \\\\ K=ab+ac+ad+bc+bd+3cd \end{cases}

Let 4 real numbers aa, bb, cc, and dd satisfy the two equations above. If the minimum and maximum values of KK are mm and M,M, respectively, what is the value of m+M? m+M ?

Give your answer to 3 decimal places.

A kk-regular graph satisfies the neighborhood diversity condition if it is possible to label each vertex with one of 1,2,3,,k1, 2, 3, \ldots, k such that for each vertex, all its neighbors have different labels.

Determine the sum of all nn satisfying 1n10001 \le n \le 1000 such that the nn-hypercube graph satisfies the neighborhood diversity condition.


Clarification:

  • In a kk-regular graph, each vertex has kk neighbors.
  • An nn-hypercube graph is the graph of an nn-dimensional hypercube. In other words, its vertices are elements of {0,1}n\{0,1\}^n (that is, nn-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.

anbn+2=(a+b)n1 a^n-b^{n+2}=(a+b)^{n-1}

aa and bb are relatively prime positive integers and n(>1)n\, (>1) is an integer. Find all solutions to the equation above and enter your answer as (a+b+n).\sum (a+b+n).


This is a generalization of the problem from 1997 Russian Olympiad:

For prime numbers pp and q,q, solve p3q5=(p+q)2.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 PP is a set of points and LL is a set of lines on a projective plane. A pair (p,l)(p, l) with pP,lLp \in P, l \in L is called an incidence if pp lies on ll. Let I(n)I(n) be the maximum number of incidences among all P,LP, L such that P=L=n|P| = |L| = n.

For example, I(4)=9I(4) = 9, achieved by having P={p1,p2,p3,p4}P = \{p_1, p_2, p_3, p_4\}, L={l1,l2,l3,l4}L = \{l_1, l_2, l_3, l_4\}, and having the set of incidences

{(p2,l1),(p3,l1),(p4,l1),(p1,l2),(p2,l2),(p1,l3),(p3,l3),(p1,l4),(p4,l4)}.\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)=Θ(nc)I(n) = \Theta(n^c) for some exponent cc; in other words, there exist constants c,N,C1,C2>0c, N, C_1, C_2 > 0 such that for all n>Nn > N, we have C1ncI(n)C2ncC_1 \cdot n^c \le I(n) \le C_2 \cdot n^c.

What is the value of 10000c?\lfloor 10000c \rfloor?


Notation: \lfloor \cdot \rfloor denotes the floor function.

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