MathDB

5

Part of 2004 IMC

Problems(2)

Problem 5 IMC 2004 Macedonia

Source:

7/25/2004
Let SS be a set of (2nn)+1\displaystyle { 2n \choose n } + 1 real numbers, where nn is an positive integer. Prove that there exists a monotone sequence {ai}1≤i≤n+2⊂S\{a_i\}_{1\leq i \leq n+2} \subset S such that ∣xi+1−x1∣≥2∣xi−x1∣, |x_{i+1} - x_1 | \geq 2 | x_i - x_1 | , for all i=2,3,…,n+1i=2,3,\ldots, n+1.
inequalitiesIMCcollege contests
Problem 11 IMC 2004 Macedonia

Source:

7/26/2004
Prove that ∫01∫01dx dy1x+∣log⁡y∣−1≤1. \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 .
integrationlogarithmsinequalitiescalculusIMCcollege contests