In a \(10 \times 10\) square, all of the cells of the upper left \(5 \times 5\) square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)
A firm recorded its expenses in pounds for 100 items, creating a list of 100 numbers (with each number having no more than two decimal places). Each accountant took a copy of the list and found an approximate amount of expenses, acting as follows. At first, he arbitrarily chose two numbers from the list, added them, discarded the sum after the decimal point (if there was anything) and recorded the result instead of the selected two numbers. With the resulting list of 99 numbers, he did the same, and so on, until there was one whole number left in the list. It turned out that in the end all the accountants ended up with different results. What is the largest number of accountants that could work in the company?
An abstract artist took a wooden \(5\times 5\times 5\) cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.
a) There is an unlimited set of cards with the words “abc”, “bca”, “cab” written. Of these, the word written is determined according to this rule. For the initial word, any card can be selected, and then on each turn to the existing word, you can either add on a card to the left or to the right, or cut the word anywhere (between the letters) and put a card there. Is it possible to make a palindrome with this method?
b) There is an unlimited set of red cards with the words “abc”, “bca”, “cab” and blue cards with the words “cba”, “acb”, “bac”. Using them, according to the same rules, a palindrome was made. Is it true that the same number of red and blue cards were used?
A cubic polynomial \(f (x)\) is given. Let’s find a group of three different numbers \((a, b, c)\) such that \(f (a)= b\), \(f (b) = c\) and \(f (c) = a\). It is known that there were eight such groups \([a_i, b_i, c_i]\), \(i = 1, 2, \dots , 8\), which contains 24 different numbers. Prove that among eight numbers of the form \(a_i + b_i + c_i\) at least three are different.
Are there functions \(p (x)\) and \(q (x)\) such that \(p (x)\) is an even function and \(p (q (x))\) is an odd function (different from identically zero)?
10 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of pasta into every other child’s bowl. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?
100 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of her pasta into other children’s bowls (to whomever she wants). What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?
A tennis tournament takes place in a sports club. The rules of this tournament are as follows. The loser of the tennis match is eliminated (there are no draws in tennis). The pair of players for the next match is determined by a coin toss. The first match is judged by an external judge, and every other match must be judged by a member of the club who did not participate in the match and did not judge earlier. Could it be that there is no one to judge the next match?
10 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of pasta into every other child’s bowl. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?