We create some segments in a regular \(n\)-gon by joining endpoints of the \(n\)-gon. What’s the maximum number of such segments while ensuring that no two segments are parallel? The segments are allowed to be sides of the \(n\)-gon - that is, joining adjacent vertices of the polygon.
At a round table, 30 people are sitting – knights and liars (knights always tell the truth, and liars always lie). It is known that each of them at that table has exactly one friend, and for each knight this friend is a liar, and for a liar this friend is a knight (friendship is always mutual). To the question “Does your friend sit next to you?” those in every other seat answered “yes”. How many of the others could also have said “Yes”?
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).
In the equality \(TIME + TICK = SPIT\), replace the same letters with the same numbers, and different letters with different digits so that the word \(TICK\) is as small as possible (there are no zeros among the digits).
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
Four children said the following about each other.
Mary: Sarah, Nathan and George solved the problem.
Sarah: Mary, Nathan and George didn’t solve the problem.
Nathan: Mary and Sarah lied.
George: Mary, Sarah and Nathan told the truth.
How many of the children actually told the truth?
Find all possible ways to represent \(2025\) as the sum of four three-digit numbers. We have the restriction that we can use only two digits across the four numbers.
The teacher wrote on the board in alphabetical order all possible \(2^n\) words consisting of \(n\) letters A or B. Then he replaced each word with a product of \(n\) factors, correcting each letter A by \(x\), and each letter B by \((1 - x)\), and added several of the first of these polynomials in \(x\). Prove that the resulting polynomial is either a constant or increasing function in \(x\) on the interval \([0, 1]\).
The graph of the function \(y=kx+b\) is shown on the diagram below. Compare \(|k|\) and \(|b|\).