Problems

The king possesses \(7\) bags of gold coins, each containing \(100\) coins. While the coins in each bag appear identical, they vary in weight and they cannot be told apart by looking. The king recalls that within these bags, one contains coins that weigh \(7\)g each, another has coins weighing \(8\)g, the third bag contains coins weighing \(9\)g, the fourth has coins weighing \(10\)g, the fifth contains coins weighing \(11\)g, the sixth holds coins weighing \(12\)g, and finally, the seventh bag contains coins weighing \(13\)g each. However, he cannot remember which bag corresponds to which coin weight.
The king reported his situation to his chancellor, pointing to one of the bags, and asked how to determine the weight of the coins in that bag. The chancellor has large two-cup scales without weights. These scales can precisely indicate whether the weights on the cups are equal or, if not, which cup is heavier. Can the chancellor ascertain which coins are in the bag indicated by the king, using no more than two weightings? The chancellor is permitted to take as many coins as necessary to conduct the weightings.

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.

A chord of a circle is a straight line between two points on the circumference of the circle. Is it possible to draw five chords on a circle in such a way that there is a pentagon and two quadrilaterals among the parts into which these chords divide the circle?

There are \(20\) chairs in the room, which come in two colors: blue and red. Each chair is occupied by either a knight or a liar. Knights always tell the truth, while liars always lie. Initially, each of those seated claimed to be sitting on a blue chair. Then, they switched seats, after which half of the participants asserted that they were now sitting on blue chairs, while the other half claimed to be sitting on red ones. How many knights are currently occupying red chairs?

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?

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 \(3\).

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?