MathDB

P6

Part of 2021/2022 Tournament of Towns

Problems(2)

Positive integer inequality

Source: 43rd International Tournament of Towns, Junior A-Level P6, Fall 2021

2/18/2023
Prove that for any positive integers a1,a2,…,ana_1, a_2, \ldots , a_n the following inequality holds true: ⌊a12a2⌋+⌊a22a3⌋+⋯+⌊an2a1⌋⩾a1+a2+⋯+an.\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n. Maxim Didin
inequalitiesalgebraTournament of Towns
43rd International Mathematics Tournament Of Towns

Source: 43rd International Mathematics Tournament Of Towns q.6

5/18/2022
There were made 7 golden, 7 silver and 7 bronze for a tournament. All the medals of the same material should weigh the same and the medals of different materials should have different weight. However, it so happened that exactly one medal had a wrong weight. If this medal is golden, it is lighter than a standard golden medal; if it is bronze, it is heavier than a standard bronze one; if it is silver, it may be lighter or heavier than a standard silver one. Is it possible to find the nonstandard one for sure, using three weighings on a balance scale with no weights?
combinatorics