Difference between revisions of "2011 USAJMO Problems/Problem 1"

Revision as of 09:51, 15 August 2021

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

Solution 1

Let $2^n + 12^n + 2011^n = x^2$ . Then $(-1)^n + 1 \equiv x^2 \pmod {3}$ . Since all perfect squares are congruent to 0 or 1 modulo 3, this means that n must be odd. Proof by Contradiction: We wish to show that the only value of $n$ that satisfies is $n = 1$ . Assume that $n \ge 2$ . Then consider the equation $2^n + 12^n = x^2 - 2011^n$ . From modulo 2, we easily know x is odd. Let $x = 2a + 1$ , where a is an integer. $2^n + 12^n = 4a^2 + 4a + 1 - 2011^n$ . Dividing by 4, $2^{n-2} + 3^n \cdot 4^{n-1} = a^2 + a + \dfrac {1}{4} (1 - 2011^n)$ . Since $n \ge 2$ , $n-2 \ge 0$ , so $2^{n-2}$ similarly, the entire LHS is an integer, and so are $a^2$ and $a$ . Thus, $\dfrac {1}{4} (1 - 2011^n)$ must be an integer. Let $\dfrac {1}{4} (1 - 2011^n) = k$ . Then we have $1- 2011^n = 4k$ . $1- 2011^n \equiv 0 \pmod {4}$ . $(-1)^n \equiv 1 \pmod {4}$ . Thus, n is even. However, it has already been shown that $n$ must be odd. This is a contradiction. Therefore, $n$ is not greater than or equal to 2, and must hence be less than 2. The only positive integer less than 2 is 1.

Solution 2

If $n = 1$ , then $2^n + 12^n + 2011^n = 2025 = 45^2$ , a perfect square.

If $n > 1$ is odd, then $2^n + 12^n + 2011^n \equiv 0 + 0 + (-1)^n \equiv 3 \pmod{4}$ .

Since all perfect squares are congruent to $0,1 \pmod{4}$ , we have that $2^n+12^n+2011^n$ is not a perfect square for odd $n > 1$ .

If $n = 2k$ is even, then $(2011^k)^2 < 2^{2k}+12^{2k}+2011^{2k}$ $= 4^k + 144^k + 2011^{2k} <$ $2011^k + 2011^k + 2011^{2k} < (2011^k+1)^2$ .

Since $(2011^k)^2 < 2^n+12^n+2011^n < (2011^k+1)^2$ , we have that $2^n+12^n+2011^n$ is not a perfect square for even $n$ .

Thus, $n = 1$ is the only positive integer for which $2^n + 12^n + 2011^n$ is a perfect square.

Solution 3

Looking at residues mod 3, we see that $n$ must be odd, since even values of $n$ leads to $2^n + 12^n + 2011^n = 2 \pmod{3}$ . Also as shown in solution 2, for $n>1$ , $n$ must be even. Hence, for $n>1$ , $n$ can neither be odd nor even. The only possible solution is then $n=1$ , which indeed works.

Solution 4

Take the whole expression mod 12. Note that the perfect squares can only be of the form 0, 1, 4 or 9 (mod 12). Note that since the problem is asking for positive integers, $12^n$ is always divisible by 12, so this will be disregarded in this process. If $n$ is even, then $2^n \equiv 4 \pmod{12}$ and $2011^n \equiv 7^n \equiv 1 \pmod {12}$ . Therefore, the sum in the problem is congruent to $5 \pmod {12}$ , which cannot be a perfect square. Now we check the case for which $n$ is an odd number greater than 1. Then $2^n \equiv 8 \pmod{12}$ and $2011^n \equiv 7^n \equiv 7 \pmod {12}$ . Therefore, this sum would be congruent to $3 \pmod {12}$ , which cannot be a perfect square. The only case we have not checked is $n=1$ . If $n=1$ , then the sum in the problem is equal to $2+12+2011=2025=45^2$ . Therefore the only possible value of $n$ such that $2^n+12^n+2011^n$ is a perfect square is $n=1$ .

Solution 5

We will first take the expression modulo $3$ . We get $2^n+12^n+2011^n \equiv -1^n+1^n \pmod 3$ .

Lemma 1: All perfect squares are equal to $0$ or $1$ modulo $3$ . We can prove this by testing the residues modulo $3$ . We have $0^2 \equiv 0 \pmod 3$ , $1^2 \equiv 1 \pmod 3$ , and $2^2 \equiv 1 \pmod 3$ , so the lemma is true.

We know that if $n$ is odd, $-1^n+1^n \equiv 0 \pmod 3$ , which satisfies the lemma's conditions. However, if $n$ is even, we get $2 \pmod 3$ , which does not satisfy the lemma's conditions. So, we can conclude that $n$ is odd.

Now, we take the original expression modulo $4$ . For right now, we will assume that $n>1$ , and test $n=1$ later. For $n>1$ , $2^n \equiv 0 \pmod 4$ , so $2^n+12^n+2011^n=-1^n \pmod 4$ .

Lemma 2: All perfect squares are equal to $0$ or $1$ modulo $4$ . We can prove this by testing the residues modulo $4$ . We have $0^2 \equiv 0 \pmod 4$ , $1^2 \equiv 1 \pmod 4$ , $2^2 \equiv 0 \pmod 4$ , and $3^2 \equiv 1 \pmod 4$ , so the lemma is true.

We know that if $n$ is even, $-1^n \equiv 0 \pmod 4$ , which satisfies the lemma's conditions. However, if $n$ is odd, $-1^n \equiv -1 \equiv 3 \pmod 4$ , which does not satisfy the lemma's conditions. Therefore, $n$ must be even.

However, a number cannot be even and odd at the same time, so this is impossible. Now, we only have to test $n=1$ . We know that $2^1+12^1+2011^1=45^2$ , so the only integer is $\boxed{n=1}$ .

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 54: / Line 54: @@
 Now, we take the original expression modulo <math>4</math>. For right now, we will assume that <math>n>1</math>, and test <math>n=1</math> later. For <math>n>1</math>, <math>2^n \equiv 0 \pmod 4</math>, so <math>2^n+12^n+2011^n=-1^n \pmod 4</math>.
-Lemma 2: All perfect squares are equal to <math>0</math> or <math>1</math> modulo <math>3</math>.
+Lemma 2: All perfect squares are equal to <math>0</math> or <math>1</math> modulo <math>4</math>.
 We can prove this by testing the residues modulo <math>4</math>. We have <math>0^2 \equiv 0 \pmod 4</math>, <math>1^2 \equiv 1 \pmod 4</math>, <math>2^2 \equiv 0 \pmod 4</math>, and <math>3^2 \equiv 1 \pmod 4</math>, so the lemma is true.

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