5. (3 points) Given the puzzle: ЛЯЛЯЛЯ + ФУФУФУ = ГГЫГЫЫР. Identical letters represent identical digits, different letters represent different digits. Find ЛЯ + ФУ.
Answer: 109

Solution: The rebus can be rewritten as GGUGUUU $=(\text{LYA} + \text{FU}) 10101$. First, this means that the last digit of $\text{LYA} + \text{FU}$ is R. Second, if $\text{LYA} + \text{FU} < 100$, the result will be a four-digit number. Let $\text{LYA} + \text{FU} = 1ZP$, where $Z$ is some digit.

Then OLAOAOY $=1ZP0000 + 1ZP00 + 1ZP$. If $P < 9$, the sixth and fourth digits of this number should match, but they do not. Therefore, $P = 9$ and the fourth digit (G) is one more than the sixth (U) due to the carry from the fifth to the fourth place.

Thus, Y + U = 9. The digit $Z$ is actually U, so $\Gamma - 1 = 0$. This means $\text{LYA} + \text{FU} = 109$.

An example exists, indeed $757575 + 343434 = 1101009$.

6. (3 points) A quadrilateral is divided into 1000 triangles. What is the smallest number of different points where the vertices of these triangles can be located?

## Answer: 503

Solution: The sum of the angles of a quadrilateral is $360^{\circ}$, and the sum of the angles of a thousand triangles is $180000^{\circ}$. Where did the extra $1796400^{\circ}$ come from? Each internal vertex, where only the angles of the triangles meet, adds $360^{\circ}$. Each vertex on the side of the original quadrilateral or on the side of one of the triangles in the division adds $180^{\circ}$. To achieve the minimum, there should only be points of the first type, and there should be $179640: 360 = 499$ of them. An example can be constructed by sequentially adding new points inside (not on the boundary) of the existing triangles.

Adding the four vertices of the original quadrilateral, we get the answer 503.