MathDB

5

Part of 2014 Tuymaada Olympiad

Problems(2)

Pairing cards such that numbers differ by 1

Source: Tuymaada 2014, Day 2, Problem 1, Senior League

7/12/2014
There is an even number of cards on a table; a positive integer is written on each card. Let aka_k be the number of cards having kk written on them. It is known that an−an−1+an−2−⋯≥0a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 for each positive integer nn. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by 11.
(A. Golovanov)
combinatorics unsolvedcombinatoricsTuymaada
Quadratic trinomials and linear function

Source: Tuymaada 2014, Day 2, Problem 1, Junior League

7/12/2014
For two quadratic trinomials P(x)P(x) and Q(x)Q(x) there is a linear function ℓ(x)\ell(x) such that P(x)=Q(ℓ(x))P(x)=Q(\ell(x)) for all real xx. How many such linear functions ℓ(x)\ell(x) can exist?
(A. Golovanov)
quadraticsfunctionalgebra solvedalgebraTuymaada