Problem 6

Let

p

be an odd prime number. How many

p

-element subsets

A

\{1,2,\ldots,2p\}

are there, the sum of whose elements is divisible by

p

Step 2 of 3: Form cyclic orbits

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In plain words

Shifting every selected first-block element preserves its cardinality and cycles after $p$ shifts.

T\mapsto T+1\pmod p

Detailed analysis

Fix the intersection with $\{p+1,\ldots,2p\}$ and shift the $r$ selected elements of $\{1,\ldots,p\}$ by $T\mapsto T+1\pmod p$ . Because $0<r<p$ , no nonzero shift fixes $T$ , so each orbit has exactly $p$ distinct subsets. Each shift increases the first-block sum by $r$ modulo $p$ .