Problems

Cambria was 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 noticed that she needed the same number of \(1\times 1\times 1\) cubes to build both the original and new cuboids. Show 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 \(CM\).

Consider a quadrilateral \(ABCD\). Choose a point \(E\) on side \(AB\). A line parallel to the diagonal \(AC\) is drawn through \(E\) and meets \(BC\) at \(F\). Then a line parallel to the other diagonal \(BD\) is drawn through \(F\) and meets \(CD\) at \(G\). And then a line parallel to the first diagonal \(AC\) is drawn through \(G\) and meets \(DA\) at \(H\). Prove the \(EH\) is parallel to the diagonal \(BD\).

Cut an arbitrary triangle into parts that can be used to build a triangle that is symmetrical to the original triangle with respect to some straight line (the pieces cannot be inverted, they can only be rotated on the plane).

The numbers from \(1\) to \(9\) are written in a row. Is it possible to write down the same numbers from \(1\) to \(9\) in a second row beneath the first row so that the sum of the two numbers in each column is an exact square?

On a Halloween night ten children with candy were standing in a row. In total, the girls and boys had equal amounts of candy. Each child gave one candy to each person on their right. After that, the girls had \(25\) more candy than they used to. How many girls are there in the row?

There are \(16\) cubes, each face of every cube is coloured yellow, black, or red (different cubes can be coloured differently). After looking at their colouring pattern, Pinoccio said that he could put all the cubes on the table in such a way that only the yellow color would be visible, on the next turn he could put the cubes in such a way that only the black color would be visible, and also he could put them in such a way that only the red color would be visible. Is there a colouring of the cubes such that he could tell the truth?

Alex writes natural numbers in a row: \(123456789101112...\) Counting from the beginning, in what places do the digits \(555\) first appear? For example, \(101\) first appears in the 10th, 11th and 12th places.