[ Rebus $]$

Authors: Vepytishev:

Solve the rebus $250 *$ LET + MGU = 2005 * YEAR. (Different letters represent different digits, and the same letters represent the same digits; some letters may represent the already existing digits 2, 5, and 0.)

a) Find at least one solution to the rebus;

b) Prove that there are no other solutions.
1) Note that for L $\leq 7$, the left side does not exceed $250 * 800+1000=201000$, while the right side is at least $2005 * 102=204510$. Therefore, L=8 or L=9.
2) If YEAR $\geq 124$, then the number on the right side is at least $2005 * 124=248620$, while the left side is at most $250 * 987+1000=247750$. Therefore, YEAR $\leq 123$ and $\Gamma=1$, which means $\mathrm{O}=0$ or $\mathrm{O}=2$.
3) The expression on the right side and the number 250 are divisible by 5, so either $\mathrm{y}=5$ or $\mathrm{V}=0$. We will show that the second case is impossible. Indeed, if $\mathrm{V}=0$, then the right side ends in zero and is therefore even, which means D is even. Since $\mathrm{O}=2$ (as O cannot be 0), the minimum value of D is 4, so

YEAR $\geq 124$. This contradicts point 2). Therefore, Y=5, which implies that D is odd.

4) There are only 3 possible values for D: D=3, D=7, and D=9. Consider the remainders when dividing the expressions on the left and right sides by 50. On the left, the remainder will be 15 (since $\Gamma=1$ and Y=5), so it must be the same on the right, which is only possible if D=3.
5) Determine the value of O. Suppose $\mathrm{O}=0$. Then on the right, we get 2005*103=206515, which means the digit T is even (otherwise, we would not get a 1 in the tens place on the left). Then LYT $\geq 824$, and $M \geq 6$ (the other digits are occupied), and the right side will be less than the left. Therefore, O=2.
6) We have YEAR=123. The case L=8 does not work (too small), leaving L=9. For the same reasons,

Note. Other reasoning (even direct guessing) is possible, but the provided solution simultaneously addresses points a) and b).

## Answer

Problem