MathDB

3

Part of 2016 Latvia National Olympiad

Problems(4)

2016 Latvia National Olympiad 3rd Round Grade9Problem3

Source:

7/22/2016
Is it possible to insert numbers 1,…,161, \ldots, 16 into a table 4×44 \times 4 (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers aa and bb satisfying\\ (a) ∣a−b∣≥6|a-b| \geq 6\\ (b) ∣a−b∣≥7|a-b| \geq 7
combinatorics
2016 Latvia National Olympiad 3rd Round Grade11Problem3

Source:

7/22/2016
Prove that for every integer nn (n>1n > 1) there exist two positive integers xx and yy (x≤yx \leq y) such that 1n=1x(x+1)+1(x+1)(x+2)+⋯+1y(y+1)\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}
algebraequationegyptian fractions
2016 Latvia National Olympiad 3rd Round Grade10Problem3

Source:

7/22/2016
Assume that real numbers xx, yy and zz satisfy x+y+z=3x + y + z = 3. Prove that xy+xz+yz≤3xy + xz + yz \leq 3.
algebrainequalities
2016 Latvia National Olympiad 3rd Round Grade12Problem3

Source:

7/22/2016
Prove that among any 18 consecutive positive 3-digit numbers, there is at least one that is divisible by the sum of its digits!
number theory