Track / Stage 6 / 359 of 400 #1359 of 1964

Problem 1359

National olympiad, first round

Combinatorics Difficulty 6.8 Find the answer

You are in a strange land and you don’t know the language. You know that ”!” and ”?” stand for addition and subtraction, but you don’t know which is which. Each of these two symbols can be written between two arguments, but for subtraction you don’t know if the left argument is subtracted from the right or vice versa. So, for instance, a?b could mean any of a − b, b − a, and a + b. You don’t know how to write any numbers, but variables and parenthesis work as usual. Given two arguments a and b, how can you write an expression that equals 20a − 18b? (12 points)

Nikolay Belukhov

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

1. Claim: We can write $0$ .

Proof: Note that the expression $(a!a)?(a!a)$ is always zero. This is because:
- If $!$ represents addition, then $a!a = a + a = 2a$ .
- If $?$ represents subtraction, then $(2a)?(2a) = 2a - 2a = 0$ .
- If $!$ represents subtraction, then $a!a = a - a = 0$ .
- If $?$ represents addition, then $(0)?(0) = 0 + 0 = 0$ .

Therefore, $(a!a)?(a!a) = 0$ regardless of the interpretation of $!$ and $?$ .

2. Claim: We can write $a + b$ and hence any linear combination of $a$ and $b$ .

Proof: Note that the expression $(a!0)!(0!b)$ is always $a + b$ . This is because:
- If $!$ represents addition, then $a!0 = a + 0 = a$ and $0!b = 0 + b = b$ . Therefore, $(a!0)!(0!b) = a + b$ .
- If $!$ represents subtraction, then $a!0 = a - 0 = a$ and $0!b = 0 - b = -b$ . Therefore, $(a!0)!(0!b) = a - (-b) = a + b$ .

Thus, we can write $a + b$ .

3. Claim: We can write $-a$ .

Proof: Note that the expression $0?((0!(a!0))?0)$ is always $-a$ . This is because:
- If $!$ represents addition, then $a!0 = a + 0 = a$ and $0!(a!0) = 0 + a = a$ . Therefore, $(0!(a!0))?0 = a - 0 = a$ if $?$ represents subtraction, or $(0!(a!0))?0 = a + 0 = a$ if $?$ represents addition.
- If $!$ represents subtraction, then $a!0 = a - 0 = a$ and $0!(a!0) = 0 - a = -a$ . Therefore, $(0!(a!0))?0 = -a - 0 = -a$ if $?$ represents subtraction, or $(0!(a!0))?0 = -a + 0 = -a$ if $?$ represents addition.

Thus, we can write $-a$ .

4. Combining the results: Since we can write $a + b$ and $-a$ , we can write any linear combination of $a$ and $b$ . Specifically, we can write $20a - 18b$ as follows:
- Write $20a$ as $20(a + 0)$ .
- Write $-18b$ as $-18(0 + b)$ .

Therefore, the expression for $20a - 18b$ can be written using the operations $!$ and $?$ .

The final answer is $\boxed{ 20a - 18b }$ .

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