Proizvolov's identity
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In mathematics, Proizvolov's identity is an identity concerning sums of differences of positive integers. The identity was posed by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads (Savchev & Andreescu 2002, p. 66).
To state the identity, take the first 2N positive integers,
- 1, 2, 3, ..., 2N − 1, 2N,
and partition them into two subsets of N numbers each. Arrange one subset in increasing order:
Arrange the other subset in decreasing order:
Then the sum
is always equal to N2.
[edit] Example
Take for example N = 3. The set of numbers is then {1, 2, 3, 4, 5, 6}. Select three numbers of this set, say 2, 3 and 5. Then the sequences A and B are:
- A1 = 2, A2 = 3, and A3 = 5;
- B1 = 6, B2 = 4, and B3 = 1.
The sum is
- | A1 − B1 | + | A2 − B2 | + | A3 − B3 | = | 2 − 6 | + | 3 − 4 | + | 5 − 1 | = 4 + 1 + 4 = 9,
which indeed equals 32.
[edit] References
- Savchev, Svetoslav & Andreescu, Titu (2002), Mathematical miniatures, vol. 43, Anneli Lax New Mathematical Library, Mathematical Association of America, ISBN 088385645X.
[edit] External links
- Proizvolov's identity at cut-the-knot.org