2021 IMO Problems/Problem 1

Problem

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

Solution

If we can guarantee that there exist $3$ cards such that every pair of them sum to a perfect square, then we can guarantee that one of the piles contains $2$ cards that sum to a perfect square. Assume the perfect squares $p^2$ , $q^2$ , and $r^2$ satisfy the following system of equations: $\usepackage{amsmath} \begin{align*} a+b &= p^2 \\ b+c &= q^2 \\ a+c &= r^2 \end{align*}$ where $a$ , $b$ , and $c$ are numbers on three of the cards. Solving for $a$ , $b$ , and $c$ in terms of $p$ , $q$ , and $r$ tells us that $a = \frac{p^2 + r^2 - q^2}{2}$ , $b=\frac{p^2 + q^2 - r^2}{2}$ , and $c=\frac{q^2 + r^2 - p^2}{2}$ . We can then substitute $p^2 = (2e-1)^2$ , $q^2 = (2e)^2$ , and $r^2 = (2e+1)^2$ to cancel out the $2$ s in the denominatior, and simplifying gives $a = 2e^2 + 1$ , $b = 2e(e-2)$ , and $c = 2e(e+2)$ . Now, we have to prove that there exists three numbers in these forms between $n$ and $2n$ when $n \ge 100$ . Notice that $b$ will always be the least of the three and $c$ will always be the greatest of the three. So it is sufficient to prove that there exists numbers in the form $2e(e-2)$ and $2e(e+2)$ between $n$ and $2n$ .

For two numbers in the form of $2e(e-2)$ and $2e(e+2)$ to be between $n$ and $2n$ , the inequalities $\usepackage{amsmath} \begin{align*} 2e(e-2) &\ge n \\ 2e(e+2) &\le 2n \\ \end{align*}$ must be satisfied. We can then expand and simplify to get that $\usepackage{amsmath} \begin{align*} e^2 - 2e - \frac{n}{2} &\ge 0 \\ e^2 + 2e - n &\le 0. \\ \end{align*}$ Then, we can complete the square on the left sides of both inequalities and isolate $e$ to get that $\usepackage{amsmath} \begin{align*} e &\ge \sqrt{1 + \frac{n}{2}} + 1 \\ e &\le \sqrt{1 + n} - 1 \\ \end{align*}$ Notice that $e$ must be an integer, so there must be an integer between $\sqrt{1 + n} - 1$ and $\sqrt{1 + \frac{n}{2}} + 1$ . If $\sqrt{1 + n} - 1$ and $\sqrt{1 + \frac{n}{2}} + 1$ differ by at least $1$ , then we can guarantee that there is an integer between them (and those integers are the possible values of $e$ ). Setting up the inequality $\sqrt{1 + n} - \sqrt{1 + \frac{n}{2}} - 2 \ge 1$ and solving for $n$ tells us that $n \in [107, \infty)$ always works. Testing the remaining $7$ numbers ( $100 - 106$ ) manually tells us that there is an integer between $\sqrt{1 + n} - 1$ and $\sqrt{1 + \frac{n}{2}} + 1$ when $n \ge 100$ . Therefore, there exists a triplet of integers $(a,b,c)$ with $a, b, c \in \{n, n+1, ..., 2n\}$ when $n \ge 100$ such that every pair of the numbers sum to a perfect square. By the pigeonhole principle, we know that $2$ of the numbers must be on cards in the same pile, and hence, when $n \ge 100$ , there will always be a pile with $2$ numbers that sum to a perfect square. $\square$

~Mathdreams

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2021 IMO Problems/Problem 1

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