Problems
On the grid paper, Theresa drew a rectangle \(199 \times 991\) with all sides on the grid lines and vertices on intersection of grid lines. How many cells of the grid paper are crossed by a diagonal of this rectangle?
The school cafeteria offers three varieties of pancakes and five different toppings. How many different pancakes with toppings can Emmanuel order? He has to have exactly one topping on each pancake.
How many six-digit numbers are there whose digits all have the same parity? That is, either all six digits are even, or all six digits are odd.
Donald’s sister Maggie goes to a nursery. One day the teacher at the nursery asked Maggie and the other children to stand a circle. When Maggie came home she told Donald that it was very funny that in the circle every child held hands with either two girls or two boys. Given that there were five boys standing in the circle, how many girls were standing in the circle?
We say that a figure is convex if a segment connecting any two points lays fully within the figure. On the picture below the pentagon on the left is convex and the one on the right is not.
Is it possible to draw \(18\) points inside a convex pentagon so that each of the ten triangles formed by its sides and diagonals contains equal amount of points?
Cambria was building various cuboids from \(1\times 1\times1\) cubes. She initially built one cuboid, then increased its length and width by \(1\) and reduced its height by \(2\). She then understood that she needs the same number of \(1\times 1\times 1\) cubes to build both the original and new cuboids. Prove that the number of cubes used for each of the cuboids is divisible by \(3\).
A labyrinth was drawn on a \(5\times 5\) grid square with an outer wall and an exit one cell wide, as well as with inner walls running along the grid lines. In the picture, we have hidden all the inner walls from you (We give you several copies to facilitate drawing) 
Please draw how the walls were arranged. Keep in mind that the numbers in the cells represent the smallest number of steps needed to exit the maze, starting from that cell. A step can be taken to any adjacent cell vertically or horizontally, but not diagonally (and only if there is no wall between them, of course).
Is it possible to cut this figure, called "camel"
a) along the grid lines;
b) not necessarily along the grid lines;
into \(3\) parts, which you can use to build a square?
(We give you several copies to facilitate drawing)
The triangle \(ABC\) is equilateral. The point \(K\) is chosen on the side \(AB\) and points \(L\) and \(M\) are on the side \(BC\) in such a way that \(L\) lies on the segment \(BM\). We have the following properties: \(KL = KM,\) \(BL = 2,\, AK = 3.\) Find the length of \(CM\).
Long ago in a galaxy far away there was a planet of liars and truth tellers, it is known that liars always tell lie, and truth tellers always respond with correct statements. All the inhabitants of the planet look identical to each other, so there is no way to distinguish between liars and truth tellers just by looking at them.
The planet is ruled by the government, where one may encounter honest governors as well as liars. The government is controlled by the High Council, where again one can meet liars and truth tellers.