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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.

Assume $a$ is a real number in $[\frac{1}{2}, \frac{2}{3}]$ . Consider two sequences $(u_n), (v_n), (n = 0, 1, \dots)$ , defined by:
$u_n = \frac{3}{2^{n+1}} \cdot (-1)^{\lfloor 2^{n+1}a \rfloor}, \quad v_n = \frac{3}{2^{n+1}} \cdot (-1)^{n+\lfloor 2^{n+1}a \rfloor}.$
a. Prove that
$\left(\sum_{i=0}^{2018} u_i\right)^2 + \left(\sum_{i=0}^{2018} v_i\right)^2 \le 72a^2 - 48a + 10 + \frac{2}{4^{2019}}.$
b. Find all value of $a$ for which equality occurs.

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Determine all sequences $a_{1}, a_{2}, \ldots$ of positive integers such that, for any pair of positive integers $m \leqslant n$ , the arithmetic and geometric means
$\frac{a_{m}+a_{m+1}+\cdots+a_{n}}{n-m+1} \quad \text{ and } \quad \left(a_{m} a_{m+1} \cdots a_{n}\right)^{\frac{1}{n-m+1}}$
are both integers.

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Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair $(a, b)$ of positive integers is called good if
$a\lceil b \nu\rceil-b\lfloor a \nu\rfloor=m$
A good pair $(a, b)$ is called excellent if neither of the pairs $(a-b, b)$ and $(a, b-a)$ is good. (As usual, by $\lfloor x\rfloor$ and $\lceil x\rceil$ we denote the integer numbers such that $x-1<\lfloor x\rfloor \leqslant x$ and $x \leqslant\lceil x\rceil<x+1$ .)
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$ .

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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 $n$ 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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Answer key — Stage 9 · Algebra

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