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Difference between revisions of "1987 IMO Problems/Problem 4"
(→‎Solution 2)
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Solution by Sawa Pavlov:
 
Solution by Sawa Pavlov:
  
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Let <math>N</math> be the set of non-negative integers. Put <math>A = N - f(N)</math> (the set of all <math>n</math> such that we cannot find <math>m</math> with <math>f(m) = n</math>). Put <math>B = f(A)</math>.  
+
Let <math>\mathbb{N}</math> be the set of non-negative integers. Put <math>A = \mathbb{N} - f(\mathbb{N})</math> (the set of all <math>n</math> such that we cannot find <math>m</math> with <math>f(m) = n</math>). Put <math>B = f(A)</math>.  
  
−
Note that <math>f</math> is injective because if <math>f(n) = f(m)</math>, then <math>f(f(n)) = f(f(m))</math> so <math>m = n</math>. We claim that <math>B = f(N) - f(f(N))</math>. Obviously <math>B</math> is a subset of <math>f(N)</math> and if <math>k</math> belongs to <math>B</math>, then it does not belong to <math>f(f(N))</math> since <math>f</math> is injective. Similarly, a member of <math>f(f(N))</math> cannot belong to <math>B</math>.  
+
Note that <math>f</math> is injective because if <math>f(n) = f(m)</math>, then <math>f(f(n)) = f(f(m))</math> so <math>m = n</math>. We claim that <math>B = f(\mathbb{N}) - f(f(\mathbb{N}))</math>. Obviously <math>B</math> is a subset of <math>f(\mathbb{N})</math> and if <math>k</math> belongs to <math>B</math>, then it does not belong to <math>f(f(\mathbb{N}))</math> since <math>f</math> is injective. Similarly, a member of <math>f(f(\mathbb{N}))</math> cannot belong to <math>B</math>.  
  
−
Clearly <math>A</math> and <math>B</math> are disjoint. They have union <math>N - f(f(N))</math> which is <math>\{0, 1, 2, \ldots , 1986\}</math>. But since <math>f</math> is injective they have the same number of elements, which is impossible since <math>\{0, 1, \ldots , 1986\}</math> has an odd number of elements.
+
Clearly <math>A</math> and <math>B</math> are disjoint. They have union <math>\mathbb{N} - f(f(\mathbb{N}))</math> which is <math>\{0, 1, 2, \ldots , 1986\}</math>. But since <math>f</math> is injective they have the same number of elements, which is impossible since <math>\{0, 1, \ldots , 1986\}</math> has an odd number of elements.
  
 
==Solution 3 ==
 
==Solution 3 ==

Revision as of 09:56, 27 May 2023

Problem

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$.

Solution 1

We prove that if $f(f(n)) = n + k$ for all $n$, where $k$ is a fixed positive integer, then $k$ must be even. If $k = 2h$, then we may take $f(n) = n + h$.

Suppose $f(m) = n$ with $m \equiv n \mod k$. Then by an easy induction on $r$ we find $f(m + kr) = n + kr$, $f(n + kr) = m + k(r+1)$. We show this leads to a contradiction. Suppose $m < n$, so $n = m + ks$ for some $s > 0$. Then $f(n) = f(m + ks) = n + ks$. But $f(n) = m + k$, so $m = n + k(s - 1) \ge n$. Contradiction. So we must have $m \ge n$, so $m = n + ks$ for some $s \ge 0$. But now $f(m + k) = f(n + k(s+1)) = m + k(s + 2)$. But $f(m + k) = n + k$, so $n = m + k(s + 1) > n$. Contradiction.

So if $f(m) = n$, then $m$ and $n$ have different residues $\pmod k$. Suppose they have $r_1$ and $r_2$ respectively. Then the same induction shows that all sufficiently large $s \equiv r_1 \pmod k$ have $f(s) \equiv r_2 \pmod k$, and that all sufficiently large $s \equiv r_2 \pmod k$ have $f(s) \equiv r_1 \pmod k$. Hence if $m$ has a different residue $r \mod k$, then $f(m)$ cannot have residue $r_1$ or $r_2$. For if $f(m)$ had residue $r_1$, then the same argument would show that all sufficiently large numbers with residue $r_1$ had $f(m) \equiv r \pmod k$. Thus the residues form pairs, so that if a number is congruent to a particular residue, then $f$ of the number is congruent to the pair of the residue. But this is impossible for $k$ odd.

Solution 2

Solution by Sawa Pavlov:

Let $\mathbb{N}$ be the set of non-negative integers. Put $A = \mathbb{N} - f(\mathbb{N})$ (the set of all $n$ such that we cannot find $m$ with $f(m) = n$). Put $B = f(A)$.

Note that $f$ is injective because if $f(n) = f(m)$, then $f(f(n)) = f(f(m))$ so $m = n$. We claim that $B = f(\mathbb{N}) - f(f(\mathbb{N}))$. Obviously $B$ is a subset of $f(\mathbb{N})$ and if $k$ belongs to $B$, then it does not belong to $f(f(\mathbb{N}))$ since $f$ is injective. Similarly, a member of $f(f(\mathbb{N}))$ cannot belong to $B$.

Clearly $A$ and $B$ are disjoint. They have union $\mathbb{N} - f(f(\mathbb{N}))$ which is $\{0, 1, 2, \ldots , 1986\}$. But since $f$ is injective they have the same number of elements, which is impossible since $\{0, 1, \ldots , 1986\}$ has an odd number of elements.

Solution 3

Consider the function $g: \mathbb{Z}_{1987} \rightarrow \mathbb{Z}_{1987}$ defined by $g(x) = f(x\; {\rm mod }\; 1987) \;{\rm mod }\; 1987$. Notice that we have $f(k) + 1987 = f(f(f(k))) = f(k + 1987)$, so that $f(x) = f(y) \;{\rm mod }\; 1987$ whenever $x = y \;{\rm mod }\; 1987$, and hence $g$ is well defined.

Now, we observe that $g$ satisfies the identity $g(g(x)) = x$, for $x \in Z_{1987}$. Thus, $g$ is an invertible function on a finite set of odd size, and hence must have a fixed point, say $a \in \mathbb{Z}_{1987}$. Identifying $a$ with its canonical representative in $\mathbb{Z}$, we therefore get $f(a) = a + 1987k$ for some non-negative integer $k$.

However, we then have $f(f(a)) = a + 1987$, while $f(f(a)) = f(a + 1987k) = 1987k + f(a) = 2\cdot1987k + a$ (where we use the identity $f(x + 1987) = f(x) + 1987$ derived above, along with $f(a) = f(a) + 1987$. However, these two equations imply that $k = \dfrac{1}{2}$, which is a contradiction since $k$ is an integer. Thus, such an $f$ cannot exist.

Note: The main step in the proof above is that the function $g$ can be shown to have a fixed point. This step works even if 1987 is replaced with any other odd number larger than 1. However, for any even number $c$, $f(x) = x + \dfrac{c}{2}$ satisfies the condition $f(f(x)) = x + c$.

--Mahamaya 21:15, 21 May 2012 (EDT)

1987 IMO (Problems) • Resources
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Problem 3 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 5
All IMO Problems and Solutions
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