MathDB

P4

Part of Russian TST 2018

Problems(2)

Generalized Moldova TST

Source: Russian TST 2018, Day 8 P4 (Groups A & B)

3/30/2023
Let a1,…,an+1a_1,\ldots,a_{n+1} be positive real numbers satisfying 1/(a1+1)+⋯+1/(an+1+1)=n1/(a_1+1)+\cdots+1/(a_{n+1}+1)=n. Prove that ∑i=1n+1∏j≠iajn⩽n+1n.\sum_{i=1}^{n+1}\prod_{j\neq i}\sqrt[n]{a_j}\leqslant\frac{n+1}{n}.
algebrainequalities
Peter is arranging his items

Source: Russian TST 2018, Day 10 P4 (Groups A & B)

3/30/2023
The natural numbers k⩾nk \geqslant n are given. Peter has nn{} objects and NN{} special ways in which he likes to lay them out in a row from left to right. He noticed that for any non-empty subset AA{} of these objects containing ∣A∣⩽k|A| \leqslant k objects, and any element a∈Aa\in A, there are exactly N/∣A∣N/|A| special ways for which element aa{} is the leftmost in the set AA{}. Prove that, under the same conditions on AA{} and aa{}, for any integer m=1,2,…,∣A∣m =1,2,\ldots,|A| there are exactly N/∣A∣N/|A| special ways for which the element aa{} is the mthm^{\text{th}} from the left in the set AA{}.
combinatorics