All Questions
5
questions
0
votes
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17
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Rewriting the Sum of indexes i and j across the ordered real numbers x(1), x(1) ... x(n) with a modulus consideration
Question is in image (sorry I don't know how to do the type set)
My attempt is halfway here and I got stuck.
Given LHS
= Sum (i=1 to i=n) for { Sum (j=1 to j=n) [x(i) - x(n)] where [u] denotes ...
9
votes
0
answers
299
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Set $S$ of integers $\ge0$ such that $\{1,2,\cdots,n\}$ partitions into $A, B$ with $\sum_\limits{a\in A}a^k=\sum_\limits{b\in B}b^k$ for all $k\in S$
Let $s_n$ be the largest size of a set $S$ of integers $\ge0$ st there exist two subsets $A,B\subseteq \{1,2,...,n\}$ that satisfy the conditions
(1) $A\cap B=\emptyset$,
(2) $A\cup B=\{1,2,...,n\}$,
(...
- 184
1
vote
2
answers
254
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Simplifying Repeated Infinite Summation
While reading a solution to a math olympiad problem, I came across a repeated, infinite summation I've never seen before. The author somehow simplifies the summation to a numerical result, but doesn't ...
- 31
2
votes
5
answers
237
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Train stops at $n$ stations
This is a problem from a mathematical contest I was unable to answer.
I am placing this problem with both the recreational math and calculus tags because I think derivatives are needed.
The problem is:...
- 3,162
19
votes
5
answers
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Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity:
$$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$
for all natural ...
- 809