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2017-07-24 Advanced

         

Consider a \( 2017 \times 2017 \) square grid. Each cell is filled in with a real number with absolute value at most 1, such that the sum of any \( 2 \times 2 \) square grid is exactly 0.

Let \(S\) be the sum of all of these real numbers. What is the maximum value of \(S\)?

Note: Below is a \(3 \times 3\) grid which satisfies the requirements above (although its sum is not necessarily maximized).

\[\Large \begin{array}{|c|c|c|} \hline -0.3 & 0.2 & -0.6 \\ \hline 0.4 & -0.3 & 0.7 \\ \hline 0.5 & -0.6 & 0.2 \\ \hline \end{array}\]

Let \(f(x)\) be a polynomial function with integer coefficients such that \[f(1) = 5, \quad f(2) = 8, \quad f(3) = 13.\]

What is the minimum possible positive value of \(f(21)\)?

On the Cartesian plane, is it possible to place an uncountable number of figure 8's (two circles that are externally tangential) that do not intersect each other?

\(\)
Note: An infinite set is uncountable if there exists no bijection to the integers.

Lazy Larry was assigned to create a bubble sort algorithm, but he didn't have the time to do it properly. He designed the following algorithm instead:

For each iteration of the algorithm:

\(\vphantom{1}\)

  • Choose two distinct elements in the list uniformly at random.

  • Exchange the placements of those elements in the list.

  • Check to see if the list is now sorted. If it is, end the algorithm. Otherwise, do another iteration.

Lazy Larry tests his algorithm on the following list:

capybara, dingo, aardvark, bandicoot

What is the expected value of the number of iterations of Larry's algorithm to alphabetize the list (A to Z)?

A knight is placed at the center of an infinitely large chess board. For large \(n,\) the number of squares on which the knight can be after exactly \(n\) moves is asymptotically \(a \cdot n^b\) for real numbers \(a\) and \(b.\) Find \(a+b.\)

Note: A chess knight moves in an "L" shape either 1 square vertically and 2 squares horizontally or 2 squares vertically and 1 square horizontally, as indicated by the stars above.

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