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Difference between revisions of "2023 INMO Problems/Problem 1"

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(2, 2), (4, 1), and (4, 4).
 
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==SOLUTION==hi
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==SOLUTION==
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To prove that there exist at least four distinct elements in <math>S</math> such that none of their pairwise products is a perfect square, we can proceed as follows:
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Each element <math>x</math> in <math>S</math> can be uniquely expressed as:
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<math>x = a^2 \cdot b</math>
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where:
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<math>a</math> is a positive integer.
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<math>b</math> is a square-free positive integer (i.e., <math>b</math> has no square factors other than 1).
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The square-free part of <math>x</math> is <math>b</math>. Two elements <math>x</math> and <math>y</math> in <math>S</math> satisfy <math>xy</math> being a perfect square if and only if they have the same square-free part.
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Based on the square-free parts, partition <math>S</math> into disjoint classes:
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<math>S = C_1 \cup C_2 \cup \dots \cup C_m</math>
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where each <math>C_i</math> consists of elements in <math>S</math> that share the same square-free part.
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The number of ordered pairs <math>(x, y)</math> such that <math>xy</math> is a perfect square is:
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<math>\sum_{i=1}^{m} |C_i|^2 = 2023</math>
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where <math>|C_i|</math> denotes the number of elements in class <math>C_i</math>.
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Using the Cauchy-Schwarz inequality:
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<math>\sum_{i=1}^{m} |C_i|^2 \geq \frac{(\sum_{i=1}^{m} |C_i|)^2}{m}</math>
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Given <math>\sum |C_i|^2 = 2023</math>, let <math>n = |S|</math>. Then:
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<math>2023 \geq \frac{n^2}{m} \implies m \geq \frac{n^2}{2023}</math>
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To maximize <math>m</math>, <math>n</math> should be as large as possible. However, even without knowing <math>n</math>, we can infer that:
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If <math>m \leq 3</math>, the inequality would impose a restriction on <math>n</math> that makes it impossible to achieve exactly 2023 ordered pairs.
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Therefore, <math>m \geq 4</math>.
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Since there are at least four distinct classes, we can select one element from each of four different classes. Let these elements be <math>a, b, c, d</math>. By construction:
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<math>a</math> and <math>b</math> belong to different classes, so <math>ab</math> is not a perfect square.
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Similarly, <math>a</math> and <math>c</math>, <math>a</math> and <math>d</math>, <math>b</math> and <math>c</math>, <math>b</math> and <math>d</math>, <math>c</math> and <math>d</math> are all in distinct classes, ensuring none of their pairwise products is a perfect square.
 +
 
 +
Thus, we have shown that there must exist at least four distinct elements in <math>S</math> such that none of their pairwise products is a perfect square.
 +
 
 +
Because <math>S</math> must contain elements from at least four distinct square-free classes, we can choose four different elements each from a separate class. Then none of their pairwise products is a square. Thus, such four distinct elements exist in <math>S</math>.

Latest revision as of 11:08, 1 October 2024

PROBLEM

Let S be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs (x, y) in S × S so that the product xy is a perfect square. Prove that one can find at least four distinct elements in S so that none of their pairwise products is a perfect square. Note: As an example, if S = {1, 2, 4}, there are exactly five such ordered pairs: (1, 1), (1, 4), (2, 2), (4, 1), and (4, 4).

SOLUTION

To prove that there exist at least four distinct elements in $S$ such that none of their pairwise products is a perfect square, we can proceed as follows:

Each element $x$ in $S$ can be uniquely expressed as:

$x = a^2 \cdot b$

where:

$a$ is a positive integer.

$b$ is a square-free positive integer (i.e., $b$ has no square factors other than 1).

The square-free part of $x$ is $b$. Two elements $x$ and $y$ in $S$ satisfy $xy$ being a perfect square if and only if they have the same square-free part.

Based on the square-free parts, partition $S$ into disjoint classes:

$S = C_1 \cup C_2 \cup \dots \cup C_m$

where each $C_i$ consists of elements in $S$ that share the same square-free part.

The number of ordered pairs $(x, y)$ such that $xy$ is a perfect square is:

$\sum_{i=1}^{m} |C_i|^2 = 2023$

where $|C_i|$ denotes the number of elements in class $C_i$.

Using the Cauchy-Schwarz inequality:

$\sum_{i=1}^{m} |C_i|^2 \geq \frac{(\sum_{i=1}^{m} |C_i|)^2}{m}$

Given $\sum |C_i|^2 = 2023$, let $n = |S|$. Then:

$2023 \geq \frac{n^2}{m} \implies m \geq \frac{n^2}{2023}$

To maximize $m$, $n$ should be as large as possible. However, even without knowing $n$, we can infer that:

If $m \leq 3$, the inequality would impose a restriction on $n$ that makes it impossible to achieve exactly 2023 ordered pairs.

Therefore, $m \geq 4$.

Since there are at least four distinct classes, we can select one element from each of four different classes. Let these elements be $a, b, c, d$. By construction:

$a$ and $b$ belong to different classes, so $ab$ is not a perfect square.

Similarly, $a$ and $c$, $a$ and $d$, $b$ and $c$, $b$ and $d$, $c$ and $d$ are all in distinct classes, ensuring none of their pairwise products is a perfect square.

Thus, we have shown that there must exist at least four distinct elements in $S$ such that none of their pairwise products is a perfect square.

Because $S$ must contain elements from at least four distinct square-free classes, we can choose four different elements each from a separate class. Then none of their pairwise products is a square. Thus, such four distinct elements exist in $S$.

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