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Stage 9 · Algebra

10 problems · IMO P2/P5; hard shortlist · mathsolympiadprep.com

The answer key prints on its own page at the end.

Determine the smallest positive integer $n$ for which there exists a polynomial
$P(X)=a_{2 n} X^{2 n}+a_{2 n-1} X^{2 n-1}+\ldots+a_{1} X+a_{0}$
with real coefficients that has both of the following properties:
- For $i=0,1, \ldots, 2 n$ we have $2014 \leq a_{i} \leq 2015$ .
- There exists a real number $\xi$ with $P(\xi)=0$ .

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Given an integer $n \ge 3$ . Let $\frac{n(n-1)}{2}$ non-negative real numbers $x_{i,j}$ ( $1 \le i < j \le n$ ) satisfy: for any $1 \le i < j < k \le n$ , we have $x_{i,j} + x_{j,k} \le x_{i,k}$ . Prove that:
$\left\lfloor \frac{n^2}{4} \right\rfloor \cdot \sum_{1 \le i < j \le n} x_{i,j}^4 \ge \left( \sum_{1 \le i < j \le n} x_{i,j}^2 \right)^2 .$

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Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ . For any $m, n \in \mathbb{Z}_{>0}$ we write $f^{n}(m)=\underbrace{f(f(\ldots f}_{n}(m) \ldots))$ . Suppose that $f$ has the following two properties:
(i) If $m, n \in \mathbb{Z}_{>0}$ , then $\frac{f^{n}(m)-m}{n} \in \mathbb{Z}_{>0}$ ;
(ii) The set $\mathbb{Z}_{>0} \backslash\left\{f(n) \mid n \in \mathbb{Z}_{>0}\right\}$ is finite.
Prove that the sequence $f(1)-1, f(2)-2, f(3)-3, \ldots$ is periodic.

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Consider the polynomial $f(x) = c x(x - 2)$ where $c$ is a positive real number. For any $n \in \mathbb{Z}^{+}$ , the notation $g_n(x)$ is a composite function $n$ times of $f$ and assume that the equation $g_n(x) = 0$ has all of the $2^n$ solutions are real numbers.
1. For $c = 5$ , find in terms of $n$ , the sum of all the solutions of $g_n(x)$ , of which each multiple (if any) is counted only once.
2. Prove that $c \ge 1$ .

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Given nonzero real numbers $\lambda_1, \lambda_2, \dots, \lambda_{2025}$ and a real number $d$ . Let $X$ be a finite set of real numbers. Define the sets:
$A = \{(x_1, \dots, x_{2025}) \in X^{2025} \mid \lambda_1 x_1 + \dots + \lambda_{2025} x_{2025} = d\};$
$B = \{(x_1, \dots, x_{2024}) \in X^{2024} \mid x_1 + \dots + x_{1012} = x_{1013} + \dots + x_{2024}\};$
$C = \{(x_1, \dots, x_{2026}) \in X^{2026} \mid x_1 + \dots + x_{1013} = x_{1014} + \dots + x_{2026}\};$
where $X^n$ denotes the set of all ordered tuples $(x_1, \dots, x_n)$ with $x_i \in X$ ( $i = 1, \dots, n$ ).
Prove: $|A|^2 \le |B| \cdot |C|$ , where $|Y|$ denotes the number of elements in the finite set $Y$ .

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Let the set of positive integers be denoted by $\mathbb{N}$ . Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ with the following property: For all positive integers $m$ and $n$ , the number $f(m)+f(n)-m n$ is different from 0 and is a divisor of the number $m f(m)+n f(n)$ .

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Let $n$ be a positive integer. Given a sequence $\varepsilon_{1}, \ldots, \varepsilon_{n-1}$ with $\varepsilon_{i}=0$ or $\varepsilon_{i}=1$ for each $i=1, \ldots, n-1$ , the sequences $a_{0}, \ldots, a_{n}$ and $b_{0}, \ldots, b_{n}$ are constructed by the following rules:
$\begin{gathered} a_{0}=b_{0}=1, \quad a_{1}=b_{1}=7 \\ a_{i+1}=\left\{\begin{array}{ll} 2 a_{i-1}+3 a_{i}, & \text{ if } \varepsilon_{i}=0, \\ 3 a_{i-1}+a_{i}, & \text{ if } \varepsilon_{i}=1, \end{array}\right. \quad \text{ for each } i=1, \ldots, n-1, \\ b_{i+1}=\left\{\begin{array}{ll} 2 b_{i-1}+3 b_{i}, & \text{ if } \varepsilon_{n-i}=0, \\ 3 b_{i-1}+b_{i}, & \text{ if } \varepsilon_{n-i}=1, \end{array}\right. \text{ for each } i=1, \ldots, n-1 \end{gathered}$
Prove that $a_{n}=b_{n}$ .

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Find the maximum positive number $M$ such that for every $n \in \mathbb{N}^*$ , there are positive numbers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ satisfying
$(a) \sum_{k=1}^{n} b_k = 1,\ 2b_k \ge b_{k-1} + b_{k+1},\ k = 2, 3, \dots, n-1,$
$(b) a_k^2 \le 1 + \sum_{i=1}^{k} a_i b_i,\ k = 1, 2, \dots, n,$
$(c) a_n = M.$

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Let $f: N \rightarrow N$ be a function from the positive integers to the positive integers for which $f(1) = 1$ , $f(2n) = f(n)$ and $f(2n+1) = f(n)+f(n+1)$ for all $n \in N$ . Prove that for any natural number $n$ , the number of odd natural numbers $m$ such that $f(m) = n$ is equal to the number of positive integers not greater than $n$ having no common prime factors with $n$ .

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Given an integer $n \ge 2$ . Find the minimum real number $\lambda$ such that for any real numbers $a_1, a_2, \dots, a_n$ , and $b$ , the following inequality holds:
$\lambda \sum_{i=1}^{n} \sqrt{|a_i - b|} + \sqrt{n \left| \sum_{i=1}^{n} a_i \right|} \ge \sum_{i=1}^{n} \sqrt{|a_i|}.$

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Answer key — Stage 9 · Algebra

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