Problems

Suppose \(x,y\) are real numbers such that \(x < y + \varepsilon\) for every \(\varepsilon > 0\). Show that \(x \leq y\).

The picture below shows a closed disc, which is just a circle with the inside filled. The grey interior represents the interior of the disc. Describe the resulting shape when you glue the circular boundary to one point.

Label the vertices of a cube with the numbers \(1,2,3,\dots,8\) so that the sum of the labels of the four vertices of each of the six faces is the same.

Is it possible to construct a 485 × 6 table with the integers from 1 to 2910 such that the sum of the 6 numbers in each row is constant, and the sum of the 485 numbers in each column is also constant?

How many ways can the numbers \(1,1,1,1,1,2,3,\dots,9\) be listed in such a way that none of the \(1\)’s are adjacent? The number 1 appears five times and each of \(2\) to \(9\) appear exactly once.

John’s local grocery store sells 7 kinds of vegetable, 7 kinds of meat, 7 kinds of grains and 7 kinds of cheese. John would like to plan the entire week’s dinners so that exactly one ingredient of each type is used per meal and no ingredients repeat during the week. How many ways can John plan the dinners?

Suppose there is an \(7 \times 7\) grid. We would like to travel from the bottom left corner to the top right corner in exactly 14 steps. A step is from one point on the grid to another point via a segment of length 1. How many paths are there? The picture below shows one possible path on the grid.

A library keeps track of its books by a code with two (not necessarily different) letters taken from A to Z, followed by a three digit number from 000 to 999. What is the maximum number of books one can keep in the library and still tell them apart by looking at their codes?

Suppose you only knew the area formula of a triangle for right-angled triangles. That is, if a base with length \(b\) and a height \(h\) of a triangle meet at a right angle, you know that the area of the triangle is \(\frac{1}{2}bh\). Can you prove the usual area formula for a general triangle?

There is a pair of parallel lines. The point \(A\) and \(B\) lie on one of the lines. The point \(C\) and \(D\) lies on the other line. We can form triangles \(\triangle ABC\) and \(\triangle ABD\). Prove that the areas of triangles \(\triangle ABC\) and \(\triangle ABD\) are equal.