Competition problem — Combinatorics

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Combinatorics Difficulty 7.3 National olympiad, round 2 Find the answer

Two robots are programmed to communicate numbers using different bases. The first robot states: "I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of which integer?

Solution

1. The first robot communicates in base 10, which is a perfect square. This is true because $10 = 10^1$ , and 10 is not a perfect square. This statement seems incorrect. However, let's assume the problem meant to say that the base 10 is used by the first robot, and it is not a perfect square.
2. The second robot communicates in base 16, which is not a perfect square. This is incorrect because $16 = 4^2$ , and 16 is indeed a perfect square. Let's assume the problem meant to say that the base 16 is used by the second robot, and it is a perfect square.
3. The sum of the bases is $10 + 16 = 26$ .
4. We need to find an integer $n$ such that $n!$ (n factorial) equals 26.

Let's calculate the factorials of integers to find the one that equals 26:
$1! = 1$
$2! = 2$
$3! = 6$
$4! = 24$
$5! = 120$

None of these factorials equal 26. Therefore, there is no integer $n$ such that $n! = 26$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.