Q1) Solve XY + YX = ZXZ where X, Y, Z are different positive digits

A1) To find the values of (X), (Y), and (Z) that satisfy the equation (XY + YX = ZXZ) where (X), (Y), and (Z) are different positive digits, we can interpret the problem as follows:

XY and YX represent two-digit numbers where (X) and (Y) are their respective tens and units digits.
ZXZ is a three-digit number where (Z) is both the hundreds and units digit, and (X) is the tens digit.
Given these interpretations, the equations can be rewritten in expanded form as:

(XY = 10X + Y)
(YX = 10Y + X)
(ZXZ = 100Z + 10X + Z = 101Z + 10X)
From the problem statement:
[ 10X + Y + 10Y + X = 101Z + 10X ]
Simplifying:
[ 11X + 11Y = 101Z + 10X ]
[ 11Y + X = 101Z ]

We're asked to find different positive digits ((1) through (9)) for (X), (Y), and (Z) which satisfy this equation. We can simplify further:
[ X + 11Y = 101Z ]
We can start by trying values for (Z) since the equation is critically dependent on (Z)'s value being feasible with (X) and (Y) as digits (i.e., between 1 and 9).
Trying (Z = 1):
[ X + 11Y = 101 ]
For (Y = 9), (X = 101 - 99 = 2). This works and all values are digits.

Trying (Z = 2):
[ X + 11Y = 202 ]
(Y = 9), (X = 202 - 99 = 103) which is not valid since (X) must be a digit.
Trying (Z = 3):
[ X + 11Y = 303 ]
Again, (Y = 9), (X = 303 - 99 = 204) which is not valid.

Continuing this pattern,
For (Z = 1) (as shown above),
[ X + 11Y = 101 ]
[ Y = 9, X = 2 ]
[ ZXZ = 101Z + 10X = 1011 + 102 = 101 + 20 = 121 ]
[ XY + YX = (102 + 9) + (109 + 2) = 29 + 92 = 121 ]
This satisfies the condition (XY + YX = ZXZ) with all variables as different positive digits.

This calculation indicates that (X = 2), (Y = 9), and (Z = 1) are the correct and only viable digits satisfying (XY + YX = ZXZ). Thus, the solution is (X = 2), (Y = 9), (Z = 1).

내가 알기로 아직 이거 답내놓는 llm은 없었던걸로 기억하는데
방금 해보니까 gpt4는 답없다고 헛소리함