Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
One of the four angles formed when two straight lines intersect is \(41^{\circ}\). What are the other three angles equal to?
In a triangle, the lengths of two of the sides are 3.14 and 0.67. Find the length of the third side if it is known that it is an integer.
a) In Wonderland, there are three cities \(A\), \(B\) and \(C\). 6 roads lead from city \(A\) to city \(B\), and 4 roads lead from city \(B\) to city \(C\). How many ways can you travel from \(A\) to \(C\)?
b) In Wonderland, another city \(D\) was built as well as several new roads – two from \(A\) to \(D\) and two from \(D\) to \(C\). In how many ways can you now get from city \(A\) to city \(C\)?
How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.
A car registration number consists of three letters of the Russian alphabet (that is, 30 letters are used) and three digits: first we have a letter, then three digits followed by two more letters. How many different car registration numbers are there?
A passenger left his things in an automatic storage room, and when he came to get his things, it turned out that he had forgotten the code. He only remembers that in the code there were the numbers 23 and 37. To open the room, you need to correctly type a five-digit number. What is the least number of codes you need to sort through in order to open the room for sure?
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.
Think of a way to finish constructing Pascal’s triangle upward.