MathDB

4

Part of 2010 Slovenia National Olympiad

Problems(4)

Find the smallest three-digit number [Slovenia 2010]

Source:

11/14/2010
Find the smallest three-digit number such that the following holds: If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.
number theory proposednumber theory
Prove that xyz+(1-x)(1-y)(1-z) ≤ 1 [Slovenia 2010]

Source:

11/14/2010
Let x,yx,y and zz be real numbers such that 0≤x,y,z≤1.0 \leq x,y,z \leq 1. Prove that xyz+(1−x)(1−y)(1−z)≤1.xyz+(1-x)(1-y)(1-z) \leq 1. When does equality hold?
probabilityfunctioninequalitiesinequalities proposed
Find all non-zero real numbers x [Slovenia 2010]

Source:

11/16/2010
Find all non-zero real numbers xx such that min⁡{4,x+4x}≥8min⁡{x,1x}.\min \left\{ 4, x+ \frac 4x \right\} \geq 8 \min \left\{ x,\frac 1x\right\} .
inequalitiesalgebra proposedalgebra
a, b, c are in an arithmetic sequence [Slovenia 2010]

Source:

11/16/2010
For real numbers a,ba, b and cc we have (2b−a)2+(2b−c)2=2(2b2−ac).(2b-a)^2 + (2b-c)^2 = 2(2b^2-ac). Prove that the numbers a,ba, b and cc are three consecutive terms in some arithmetic sequence.
arithmetic sequencealgebra proposedalgebra