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Stage 9 · Algebra

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

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1. 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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2. 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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3. Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called simple if it satisfies $(x \times y) \times z = x \times (y \times z)$ for all $x, y, z \in S$ and $x \times y \in \{x, y\}$ for all $x, y \in S$. Given a simple operation $\times$ on $S$, any string of elements in $S$ can be reduced to a single element, such as $xyz \rightarrow x \times (y \times z)$. A string of variables in $S$ is called full if it contains each variable in $S$ at least once, and two strings are equivalent if they evaluate to the same variable regardless of which simple $\times$ is chosen. For example, $xxx$, $xx$, and $x$ are equivalent, but these are only full if $n = 1$. Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$. Determine the number of elements of $T$.
   (This problem was suggested by John Berman.)

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4. 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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5. Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a, b)$, $f(b, c)$, $f(c, a)$ equals the median of $a, b, c$. (The median of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in non-decreasing order.)

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6. 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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7. Let $S$ be a proper subset of $\mathbb{R}$ (i.e. $S \neq \mathbb{R}$) having at least two elements. Suppose there exists a function $f: \mathbb{R} \to \mathbb{R}$ satisfying the following conditions:
   (i) $f(a + x + y) + f(f(a)) + f(x) + f(y) = x + y$; and
   (ii) $f(axy) + f(a) + f(x)f(y) = xy$
   for any real numbers $a \notin S$ and $x, y \in S$. Find all such function(s) $f$.

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8. Let $m, n \in \mathbb{N}^*, m, n > 1, a_{ij}$ ($i = 1, 2, \dots, n$, $j = 1, 2, \dots, m$) be non-negative real numbers (not all zero). Find the maximum and minimum values of
   $$f = \frac{n \sum_{i=1}^{n} (\sum_{j=1}^{m} a_{ij})^2 + m \sum_{j=1}^{m} (\sum_{i=1}^{n} a_{ij})^2}{(\sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij})^2 + mn \sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij}^2}.$$

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9. Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right)$.
   Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.

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10. We say that a function $f: \mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form
    $$f\left(x_1, \ldots, x_k\right)=\max_{i=1, \ldots, m} \min_{j=1, \ldots, n} P_{i, j}\left(x_1, \ldots, x_k\right)$$
    where $P_{i, j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.

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Answer key — Stage 9 · Algebra

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