# Problems of the Week

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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.

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

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 $$m+M ?$$

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.


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 $$\lfloor 10000c \rfloor?$$


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

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