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Part of 2011 Middle European Mathematical Olympiad

Problems(2)

At least ceil[2n/9] of the languages can be chosen

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-4

9/6/2011
Let n≥3n \geq 3 be an integer. At a MEMO-like competition, there are 3n3n participants, there are n languages spoken, and each participant speaks exactly three different languages. Prove that at least ⌈2n9⌉\left\lceil\frac{2n}{9}\right\rceil of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.
Note. ⌈x⌉\lceil x\rceil is the smallest integer which is greater than or equal to xx.
ceiling functionfloor functiongraph theoryexpected valuecombinatorics proposedcombinatoricsProbabilistic Method
If k^3 - m^3 | km(k^2 - m^2), show that (k - m)^3 > 3km

Source: Middle European Mathematical Olympiad 2011 - Individuals I-4

9/6/2011
Let kk and mm, with k>mk > m, be positive integers such that the number km(k2−m2)km(k^2 - m^2) is divisible by k3−m3k^3 - m^3. Prove that (k−m)3>3km(k - m)^3 > 3km.
number theory proposednumber theory