Problems

All of the rectangles in the figure below, which is drawn to scale, are similar to the big rectangle (that is, their sides are in the same ratio). Each number represents the area of the rectangle. What is the length \(AB\)?

For any positive integer \(k\), the factorial \(k!\) is defined as a product of all integers between 1 and \(k\) inclusive: \(k! = k \times (k-1) \times ... \times 1\). What’s the remainder when \(2025!+2024!+2023!+...+3!+2!+1!\) is divided by \(8\)?

Find all functions \(f\) from the real numbers to the real numbers such that \(xy=f(x)f(y)-f(x+y)\) for all real numbers x and y.

There are two imposters and seven crewmates on the rocket ‘Plus’. How many ways are there for the nine people to split into three groups of three, such that each group has at least two crewmates? The two imposters and seven crewmates are all distinguishable from each other, but we’re not concerned with the order of the three groups.

For example: \(\{I_1,C_1,C_2\}\), \(\{I_2,C_3,C_4\}\) and \(\{C_5,C_6,C_7\}\) is the same as \(\{C_3,C_4,I_2\}\), \(\{C_5,C_6,C_7\}\) and \(\{I_1,C_2,C_1\}\) but different from \(\{I_2,C_1,C_2\}\), \(\{I_1,C_3,C_4\}\) and \(\{C_5,C_6,C_7\}\).

Draw the plane tiling with regular hexagons.

Let \(x\), \(y\), \(z\) and \(w\) be non-negative integers. Find all solutions to \(2^x3^y-5^z7^w=1\).

Can you tile the plane with regular octagons?

Draw how to tile the whole plane with figures, composed from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), and \(5\times 5\) where squares of all sizes are used the same amount of times in the design of the figure.

Let \(a,b,c >0\) be positive real numbers with \(abc \leq 1\). Prove that \[\frac{a}{c} + \frac{b}{a} + \frac{c}{b} \geq a+b+c.\]

From IMO 1999. Let \(n\geq 2\) be an integer. Determine the least possible constant \(C\) such that the inequality \[\sum_{1\leq i<j\leq n} x_ix_j(x_i^2 + x_j^2) \leq C(\sum_{1\leq i\leq n}x_i)^4\] holds for all non-negative real numbers \(x_i\). For this constant \(C\) find out when the equality holds.