Pigeonhole Principle

Pigeonhole Principle: Level 4 Challenges


If we pick 77 numbers randomly from the set {1,2,3,4,...,150}\{1,2,3,4,...,150\}, we are guaranteed to have at least kk pairs of numbers where the difference between each pair is 19. What is the maximum possible value of k?k?

Please treat a pair as a combination, not a permutation.

From the set {1,2,3,...,n1,2,3,...,n}, 1010 numbers are removed such that there is no Arithmetic Progression of length 1111 among the numbers left in the set.

Find the smallest nn such that no matter which 1010 numbers are removed, there always is one Arithmetic Progression of length 1111.

A 1000×10001000\times1000 array is to be filled with integers between 1 and 1000 inclusive, with each integer appearing 1000 times. What is the maximum value of nn such that there necessarily exists a row or column with at least nn distinct integers?

Let a0,a1,,a7a_0, a_1, \cdots, a_7 be any 88 distinct integers. Let PP be the product of their pairwise differences, that is:

P=i<j(aiaj)P = \prod _ {i < j} {(a_i - a_j)}

What is the greatest integer which always divides P?P?

Determine the least positive integer n3 n \geq 3 for which the following condition holds: No matter how the elements of the set of the first nn positive integers, i.e. {1,2,n} \{1, 2, \ldots n \}, are colored in red or blue, there are (not necessarily distinct) integers x,y,z x, y, z, and w w in a set of the same color such that x+y+z=w x + y + z = w.

Details and assumptions

The phrase not necessarily distinct means that the integers can be repeated. For example, if 1,2,41, 2, 4 are all colored red, then we have 1+1+2=4 1 + 1 + 2 = 4 which would satisfy the condition.


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