Problems

Let \(ABC\) be a triangle. Prove that the heights \(AD\), \(BE\), \(CF\) intersect in one point.

Let \(ABC\) be a triangle. Prove that the medians \(AD\), \(BE\), \(CF\) intersect in one point.

Let \(ABC\) be a triangle with medians \(AD\), \(BE\), \(CF\). Prove that the triangles \(ABC\) and \(DEF\) are similar. What is their similarity coefficient?

How many subsets are there of \(\{1,2,...,10\}\) (the integers from \(1\) to \(10\) inclusive) containing no consecutive digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).
For example, when \(n=3\), we have \(8\) subsets overall but only \(5\) contain no consecutive integers. The \(8\) subsets are \(\varnothing\) (the empty set), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,3\}\), \(\{1,2\}\), \(\{2,3\}\) and \(\{1,2,3\}\), but we exclude the final three of these.

Dario is making a pizza. He has the option to choose from \(3\) different types of flatbread, \(4\) different types of cheese and \(2\) different sauces. How many different pizzas can he make?

Determine the number of \(4\)-digit numbers that are composed entirely of distinct even digits.

There are \(10\) boys that need to be arranged in a line. Two of these boys are brothers, who need to have an even number of other boys between them. How many possible arrangements are there?

Steve, a student, has discovered that he lost most of his socks, and as a result, none of them match anymore. He has \(4\) black right socks, \(6\) blue right socks, \(8\) black left socks, and \(5\) blue left socks. Additionally, he has \(2\) pairs of blue trousers and \(3\) pairs of black trousers. Steve wants to ensure that his clothing items match in colour, so they desire to have left socks, right socks, and trousers of the same colour. How many different ways can Steve dress up given these conditions?

Determine the number of \(5\)-digit numbers that have only one odd digit and all other digits are even and distinct.

A group of \(4\) adults and \(5\) children is on a mountain hiking trip. At one point, the path becomes really narrow, and the hikers have to move in a line. They agreed that the line has to both start and end with an adult, for safety reasons. In how many ways can they arrange themselves?