Problems

Let \(ABCD\) be a quadrilateral with the point \(E\) of intersection of diagonals. Consider the triangles \(ABE\), \(BCE\), \(CDE\), \(ADE\) and their correspondent points of intersection of medians \(K,J,M,L\).

Prove that \(KJML\) is a parallelogram.

At the Oscar Awards 2025, 5 films were nominated for Best Production Design and 5 films were nominated for Best Cinematography. In fact, 3 films were nominated for both categories. What is the total number of films nominated for these two categories?

150 young adults were asked how they commuted to work. 125 said they took the underground and 93 said they cycled. Of all the people interviewed, 72 said they both cycled and took the underground. How many people do not cycle and do not take the underground?

Show that there are no more than 269 prime numbers less than or equal to 1000.

200 people were asked if they drank one of the following beverages regularly: tea, coffee and beer. 165 people said they drank at least one of these beverages. Funnily, for every choice of a pair of beverages, exactly 122 people said they drank at least one beverage out of the pair. The even stranger fact was that for each choice of a beverage, exactly 73 people admitted to drinking it.

How many people drink all three beverages?

Each student chooses at least one from the \(n\) different modules offered at a university. Let us number these modules as \(1,2,3,\dots,n\). For each natural number \(1\leq k\leq n\), we denote the number of students choosing the modules \(i_1,\dots,i_k\) by \(S(i_1,\dots,i_k)\). Give a formula for the number of students in terms of the numbers \(S(i_1,\dots,i_k)\).

As an example, if \(n = 5\), \(k=3\) and we look at \(i_1 = 4,i_2 = 2, i_3 =1\), then \(S(i_1,i_2,i_3) = S(4,2,1)\) is the number of students picking the modules \(1,2,4\).

There are \(n\) seats on a plane and each of the \(n\) passengers sat in the wrong seat. What is the total number of ways this could happen?

Let \(n\geq 2\) be an integer. Fix \(2n\) points in space, so that no four points lie on a common plane. Suppose there are \(n^2+1\) segments between these points. Show that these segments must form at least \(n\) triangles.

Elections are approaching in Problemland! There are three candidates for president: \(A\), \(B\), and \(C\).

An opinion poll reports that \(65\%\) of voters would be satisfied with \(A\), \(57\%\) with \(B\), and \(58\%\) with \(C\). It also says that \(28\%\) would accept \(A\) or \(B\), \(30\%\) \(A\) or \(C\), \(27\%\) \(B\) or \(C\), and that \(12\%\) would be content with all three candidates.

Show that there must have been a mistake in the poll.

You are creating passwords of length \(8\) using only the letters \(A\), \(B\), and \(C\). Each password must use all three letters at least once.

How many such passwords are there?