MathDB

2011 Germany Team Selection Test

Part of Germany Team Selection Test

Subcontests

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Inequality in Functional Equation

We call a function f:Q+→Q+f: \mathbb{Q}^+ \to \mathbb{Q}^+ good if for all x,y∈Q+x,y \in \mathbb{Q}^+ we have: f(x)+f(y)≥4f(x+y).f(x)+f(y)\geq 4f(x+y). a) Prove that for all good functions f:Q+→Q+f: \mathbb{Q}^+ \to \mathbb{Q}^+ and x,y,z∈Q+x,y,z \in \mathbb{Q}^+ f(x)+f(y)+f(z)≥8f(x+y+z)f(x)+f(y)+f(z) \geq 8f(x+y+z) b) Does there exists a good functions f:Q+→Q+f: \mathbb{Q}^+ \to \mathbb{Q}^+ and x,y,z∈Q+x,y,z \in \mathbb{Q}^+ such that f(x)+f(y)+f(z)<9f(x+y+z)?f(x)+f(y)+f(z) < 9f(x+y+z) ?

Numbers on Vertices of A Regular n-gon

Vertices and Edges of a regular nn-gon are numbered 1,2,…,n1,2,\dots,n clockwise such that edge ii lies between vertices i,i+1mod  ni,i+1 \mod n. Now non-negative integers (e1,e2,…,en)(e_1,e_2,\dots,e_n) are assigned to corresponding edges and non-negative integers (k1,k2,…,kn)(k_1,k_2,\dots,k_n) are assigned to corresponding vertices such that: ii) (e1,e2,…,en)(e_1,e_2,\dots,e_n) is a permutation of (k1,k2,…,kn)(k_1,k_2,\dots,k_n). iiii) ki=∣ei+1−ei∣k_i=|e_{i+1}-e_i| indexesmod  n\mod n.
a) Prove that for all n≥3n\geq 3 such non-zero nn-tuples exist. b) Determine for each mm the smallest positive integer nn such that there is an nn-tuples stisfying the above conditions and also {e1,e2,…,en}\{e_1,e_2,\dots,e_n\} contains all 0,1,2,…m0,1,2,\dots m.
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