One day a strange notebook was found on the stairs. It contained one hundred statements:
“There is exactly one incorrect statement in this notebook”;
“There are exactly two incorrect statements in this notebook”;
“There are exactly three incorrect statements in this notebook”;
...
“There are exactly one hundred incorrect statements in this notebook.”
Are any of these statements true, and if so, which ones?
Jack the goldminer extracted 9 kg of golden sand. Will he be able to measure 2 kg of sand in three goes with the help of scales: a) with two weights of 200 g and 50 g; b) with one weight of 200 g?
Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.
a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.
b) Will this statement remain true if instead of the difference we considered the total?
In the gymnasium, all students know at least one of the ancient languages – Greek or Latin, some – both languages. 85% of all children know the Greek language and 75% know Latin. How many students know both languages?
Two classes with the same number of students took a test. Having checked the test, the strict teacher Mr Jones said that he gave out 13 more twos than other marks (where the marks range from 2 to 5 and 5 is the highest). Was Mr Jones right?
Before the start of the Olympics, the price of hockey pucks went up by 10%, and after the end of the Olympics they fell by 10%.
When were the pucks more expensive – before the price rise or after the fall?
What word is encrypted: 22212221265121? Each letter is replaced by its number in the English alphabet.
Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?
Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?