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3

Part of 2006 Poland - Second Round

Problems(2)

classical - symmetrical with 3 variables

Source: Polish second round 2006

2/24/2006
Positive reals a,b,ca,b,c satisfy ab+bc+ca=abcab+bc+ca=abc. Prove that: a4+b4ab(a3+b3)+b4+c4bc(b3+c3)+c4+a4ca(c3+a3)≥1\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1
inequalitiesinequalities proposed
largest possible cardinality of a set

Source:

2/25/2006
Given is a prime number pp and natural nn such that p≥n≥3p \geq n \geq 3. Set AA is made of sequences of lenght nn with elements from the set {0,1,2,...,p−1}\{0,1,2,...,p-1\} and have the following property: For arbitrary two sequence (x1,...,xn)(x_1,...,x_n) and (y1,...,yn)(y_1,...,y_n) from the set AA there exist three different numbers k,l,mk,l,m such that: xk≠ykx_k \not = y_k, xl≠ylx_l \not = y_l, xm≠ymx_m \not = y_m. Find the largest possible cardinality of AA.
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