MathDB

A7

Part of 2024-IMOC

Problems(1)

REAL functions

Source: 2024 IMOC A7 (Night 4)

8/8/2024
Given positive integers nn, P1P_1, P2P_2, …PnP_n and two sets B={(a1,a2,…,an)∣ai=0∨1, ∀i∈N},S={(x1,x2,…,xn)∣1≤xi≤Pi∧xi∈N, ∀i∈N}B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \} A function f:S→Zf:S \to \mathbb{Z} is called Real, if and only if for any positive integers (y1,y2,…,yn)(y_1,y_2,…,y_n) and positive integer aa which satisfied 1≤yi≤Pi−a 1 \leq y_i \leq P_i-a ∀i∈N\forall i \in \mathbb{N}, we always have: \begin{align*} \sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\ \sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&. \end{align*} Find the minimum of ∑i1=1P1∑i2=1P2....∑in=1Pn∣f(i1,i2,...,in)∣\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|, where ff is a Real function.
Proposed by tob8y
algebrafunctionReal