Problems

All the positive fractions smaller than $1$ with denominators not more than $100$ are written in a row. Isley and Ella put signs $"+"$ or $"-"$ in front of any fraction, which does not yet have a sign before it. They write signs in turns, but it is known that Isley has to make the last move and calculate the resulting sum. If the total sum turns out to be an integer number, then Ella will give her a chocolate bar. Will Isley be able to get a chocolate bar regardless of Ella’s actions?

Abigail’s little brother Carson found a big rectangular cake in the fridge and cut a small rectangular piece out of it.

Now Abigail needs to find a way to cut the remaining cake into two pieces of equal area with only one straight cut. How could she do that? The removed piece can be of any size or orientation.

Is it possible to cut an equilateral triangle into three equal hexagons?

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 $1\times 1\times 1$ 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 $3$.

Michael used different numbers $\{0,1,2,3,4,5,6,7,8,9\}$ to put in the circles in the picture below, without using any one of them twice. Inside each triangle he wrote down either the sum or the product of the numbers at its vertices. Then he erased the numbers in the circles. Which numbers need to be written in circles so that the condition is satisfied?

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 $CM$.

Solve the puzzle: $$\text{AC}\times \text{CC}\times \text{K}=2002.$$ Different letters correspond to different digits, identical letters correspond to identical digits.