Prove that for every non-negative integer $ n, $ there exists a non-negative integer $ m $ such that $$ \left( 1+\sqrt{2} \right)^n=\sqrt m +\sqrt{m+1} . $$

Find all natural numbers $ n,x,y $ such that $ \big| 2^x-n^{y+1}\big| =1 . $

Let $ ABC $ be a triangle and $ A_1,B_1,C_1 $ the projections of $ A,B,C $ on their opposite sides. Let $ A_2,A_3 $ be the projection of $ A_1 $ on $ AB, $ respectively on $ AC. B_2,B_3,C_2,C_3 $ are defined analogously. Moreover, $ A_4 $ is the intersection of $ B_2B_3 $ with $ C_2C_3; B_4, $ the intersection of $C_2C_3 $ with $ A_2A_3; C_4, $ the intersection of $ A_2A_3 $ with $ B_2B_3. $ Show that $ AA_4,BB_4 $ and $ CC_4 $ are concurrent.

At a table tennis tournament, each one of the $ n\ge 2 $ participants play with all the others exactly once. Show that, at the end of the tournament, one and only one of these propositions will be true: $ \text{(1)} $ The players can be labeled with the numbers $ 1,2,...,n, $ such that $ 1 $ won $ 2, 2 $ won $ 3,...,n-1 $ won $ n $ and $ n $ won $ 1. $ $ \text{(2)} $ The players can be partitioned in two nonempty subsets $ A,B, $ such that whichever one from $ A $ won all that are in $ B. $

Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies: $ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $ $ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $ $ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $

Let be a structure formed by $ n\ge 4 $ points in space, four by four noncoplanar, and two by two connected by a wire. If we cut the $ n-1 $ wires that connect a point to the others, the remaining point is said to be isolated. The structure is said to be disconnected if there are at least two points for which there isn´t a chain of wires connecting them. So, initially, it´s not disconnected. $ \text{(1)} $ Prove that, by cutting a number smaller or equal with $ n-2, $ the structure won´t become disconnected. $ \text{(2)} $ Determine the minimum number of wires that needs to be cut so that the remaining structure is disconnected, yet every point, not isloated.

Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $ Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $

Show that any subset of $ A=\{ 1,2,...,2007\} $ having $ 27 $ elements contains three distinct numbers such that the greatest common divisor of two of them divides the other one. Dan Schwarz