MathDB

2

Part of 2014 International Zhautykov Olympiad

Problems(2)

International Zhautykov olympiad 2014 problem 2

Source:

1/14/2014
Does there exist a function f:R→Rf: \mathbb R \to \mathbb R satisfying the following conditions: (i) for each real yy there is a real xx such that f(x)=yf(x)=y , and (ii) f(f(x))=(x−1)f(x)+2f(f(x)) = (x - 1)f(x) + 2 for all real xx ?
Proposed by Igor I. Voronovich, Belarus
functionalgebrafunctional equationalgebra proposed
Sets

Source:

1/15/2014
Let U={1,2,…,2014}U=\{1, 2,\ldots, 2014\}. For positive integers aa, bb, cc we denote by f(a,b,c)f(a, b, c) the number of ordered 6-tuples of sets (X1,X2,X3,Y1,Y2,Y3)(X_1,X_2,X_3,Y_1,Y_2,Y_3) satisfying the following conditions: (i) Y1⊆X1⊆UY_1 \subseteq X_1 \subseteq U and ∣X1∣=a|X_1|=a; (ii) Y2⊆X2⊆U∖Y1Y_2 \subseteq X_2 \subseteq U\setminus Y_1 and ∣X2∣=b|X_2|=b; (iii) Y3⊆X3⊆U∖(Y1∪Y2)Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2) and ∣X3∣=c|X_3|=c. Prove that f(a,b,c)f(a,b,c) does not change when aa, bb, cc are rearranged.
Proposed by Damir A. Yeliussizov, Kazakhstan
symmetrycombinatorics unsolvedcombinatorics