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Stage 10 · Algebra

8 problems · Hardest shortlist tier · mathsolympiadprep.com

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1. Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exist $n$ positive integers $x_{1}, x_{2}, \ldots, x_{n}$ such that $\frac{1}{2}<\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}<2$ and $\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leqslant i, j \leqslant n$.

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2. An integer $n \geqslant 3$ is given. We call an $n$-tuple of real numbers $(x_{1}, x_{2}, \ldots, x_{n})$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have
   $$\sum_{i=1}^{n-1} y_{i} y_{i+1} = y_{1} y_{2} + y_{2} y_{3} + y_{3} y_{4} + \cdots + y_{n-1} y_{n} \geqslant -1.$$
   Find the largest constant $K = K(n)$ such that
   $$\sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j} \geqslant K$$
   holds for every Shiny $n$-tuple $(x_{1}, x_{2}, \ldots, x_{n})$.

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3. Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
   - In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
   - In the second line, Gugu writes down every number of the form $q a b$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line.
   - In the third line, Gugu writes down every number of the form $a^{2}+b^{2}-c^{2}-d^{2}$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
   Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

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4. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
   $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a))=f\left(a b^{2}+b c^{2}+c a^{2}\right)-f\left(a^{2} b+b^{2} c+c^{2} a\right)$$
   for all real numbers $a, b, c$.

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5. At the beginning we have a triple of pairwise distinct positive integers $(a, b, c)$ which satisfies $a + b + c = 2015$. Then in each step we replace the current triple of numbers $(x, y, z)$ with the triple $(y + z - x, z + x - y, x + y - z)$. At least how many steps must we make so that starting with a triple $(a, b, c)$ we will certainly get a triple which has at least one negative number?

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6. Let $N$ be a positive integer and let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers. Suppose that, for each $n > N$, $a_{n}$ is equal to the number of times $a_{n-1}$ appears in the list $a_{1}, a_{2}, \ldots, a_{n-1}$.
   Prove that at least one of the sequences $a_{1}, a_{3}, a_{5}, \ldots$ and $a_{2}, a_{4}, a_{6}, \ldots$ is eventually periodic.

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7. Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that
   $$f(xy + x + y) + f\left(\frac{1}{x}\right) f\left(\frac{1}{y}\right) = 1$$

   $$f(xy + x + y) + f\left(\frac{1}{x}\right) f\left(\frac{1}{y}\right) = 1 \quad \text{for every } x, y \in \mathbb{R}_+.$$

   holds for all $x, y \in \mathbb{R}_+$.

   Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that
   $$f(xy + x + y) + f\left(\frac{1}{x}\right) f\left(\frac{1}{y}\right) = 1$$
   for every $x, y \in \mathbb{R}_+$.

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8. Let $\mathbb{Q}$ be the set of rational numbers. Let $f: \mathbb{Q} \rightarrow \mathbb{Q}$ be a function such that the following property holds: for all $x, y \in \mathbb{Q}$,
   $$f(x+f(y))=f(x)+y \quad \text{or} \quad f(f(x)+y)=x+f(y).$$

   Determine the maximum possible number of elements of $\{f(x)+f(-x) \mid x \in \mathbb{Q}\}$.

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Answer key — Stage 10 · Algebra

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