2
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1answer
227 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
0
votes
1answer
69 views

Pigeonhole proof of the existence of two numbers with given sum [duplicate]

Let $|W|=m+1$ and $W$ be a subset of $X=\{1,2,3,\dots ,2m\}$ ($m$ is any natural number). Prove there exists two numbers in $W$ whose sum is $2m+1$. Can anyone give me a hint to prove this? I ...
0
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1answer
82 views

Perfect Fourth Power - Pigeon Hole Principle

Let $a_1, a_2, ..., a_n$ be positive integers all of whose prime divisors are $\le$ 13. Show that if $n \ge 193$ then there exists four of these integers whose product is a perfect fourth power. I ...
0
votes
2answers
61 views

Pigeonhole question and generalization

Let H be a regular hexagon with side length 1 unit. (a) Show that if more than 6 points are speci ed inside H then the points of at least one pair of them are at most 1 unit apart. (b) State and ...
2
votes
2answers
93 views

Pigeonhole Principle Exercise

Show that any subset of $\{1, 2, 3, ..., 200\}$ having more than $100$ members must contain at least one pair of integers which add to $201$. I think it is doable using the Pigeonhole Principle.
0
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1answer
58 views

Pigeonhole Principle to solve question straightforward

A store wants to celebrate its anniversary and will give a $200 shopping certi cate to the first customer to enter the store whose birthday is the same as that of two other previously admitted ...
3
votes
1answer
79 views

Pigeonhole Proof

Let $n_1,n_2,\ldots,n_t$ be positive integers. Show that if $n_1+n_2+\cdots+n_t-t+1$ objects are placed into $t$ boxes, then for some $i$, $i = 1,2,\ldots,t$, the $i$th box contatins at least $n_i$ ...
0
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1answer
534 views

pigeonhole fun discrete math

How do i use the pigeon hole principle for these questions? A drawer contains 6 pairs of black, 5 pairs of white, 5 pairs of red, and 4 pairs of green socks. (a) How many single socks do we have to ...