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Board problem with complex numbers
egxa   1
N Apr 18, 2025 by hectorraul
Source: All Russian 2025 11.1
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[ a^2 + b^2 + 1 = 2ab \]Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[ c^2 + d^2 + 2025 = 2cd \]
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egxa
Apr 18, 2025
hectorraul
Apr 18, 2025
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Board problem with complex numbers
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Source: All Russian 2025 11.1
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egxa
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egxa
#1 h  Apr 18, 2025, 9:45 AM • 1 Y
Y by Tung-CHL
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[ a^2 + b^2 + 1 = 2ab \]Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[ c^2 + d^2 + 2025 = 2cd \]
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hectorraul
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hectorraul
#2 h  Apr 18, 2025, 11:34 AM • 1 Y
Y by Quidditch
Consider the graph $G$ of $777$ nodes, one for each complex number and draw edges if the condition is held.

$\textbf{claim:}$ $G$ is form by exactly $17$ disconnected and simple path.
$\textbf{proof:}$ Condition is equivalent to $a-b = \pm i$ meaning that $a$ can only be connected to $a-i$ and $a+i$, this ensure that $G$ is a union of disconnected and simple path.
Suppose that we have $k$ of these path, each of length $l_1,l_2,...,l_k$. The we have that $\sum l_i = 760$ and $777 = \sum (l_i+1) = \sum l_i + k = 760+k$, then $k = 17$.$\square$

Now, by PHP one of these chains has at least $\lceil 777/17\rceil = 46$ nodes. So we have a complex number $c$ and $c+45i$ and its easy to check that
\[ c^2+(c+45i)^2+2025 = 2c(c+45i) \].
This post has been edited 4 times. Last edited by hectorraul, Apr 20, 2025, 6:25 PM
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