Prove that

\(\displaystyle \left(1-\frac14\right) \left(1-{1\over 9}\right) \ldots \left(1-{1\over n^2}\right) = {n+1\over 2n} \)

Difficulty: 3. Solution is available for this problem.

Prove that the following identity holds for all positive integer \(\displaystyle n\):

\(\displaystyle \frac{1}{1\cdot 3} +\frac{1}{3\cdot 5}+\ldots + \frac{1}{(2n-1)\cdot (2n+1)}=\frac{n}{2n+1}.\)

Difficulty: 3. Solution is available for this problem.

Prove that the following identity holds for all positive integer \(\displaystyle n\):

\(\displaystyle \frac{x^n -y^n}{x-y}=x^{n-1}+x^{n-2}\cdot y +\ldots +x\cdot y^{n-2} + y^{n-1} \)

Difficulty: 3.

Prove that the following identity holds for all positive integer \(\displaystyle n\):

\(\displaystyle 1^3 +\ldots +n^3 =\left( \frac{n\cdot (n+1)}{2} \right) ^2 .\)

Difficulty: 3. Solution is available for this problem.

Prove that the following identities hols for all positive integer \(\displaystyle n\):
 (1)  \(\displaystyle \displaystyle 1-\frac{1}{2} +\frac{1}{3} -\ldots -\frac{1}{2n} =\frac{1}{n+1} + \ldots +\frac{1}{2n};\)
 (2)  \(\displaystyle \displaystyle \frac{1}{1\cdot 2} +\ldots + \frac{1}{(n-1)\cdot n}=\frac{n-1}{n} .\)

Difficulty: 3.

Prove that \(\displaystyle 1 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \dots +n (3n + 1) = n (n + 1)^2\).

Difficulty: 3.

 (1)  Let \(\displaystyle a_1 =1\) and \(\displaystyle a_{n+1} =\sqrt{2a_n +3}\). Prove that \(\displaystyle \forall n\in \N \ a_n \le a_{n+1}\).
 (2)  Let \(\displaystyle a_1 =0.9\) and \(\displaystyle a_{n+1} =a_n -a_n^2 \). Prove that \(\displaystyle \forall n\in \N\ a_{n+1} <a_n \;\) and \(\displaystyle \;0<a_n <1\).

Difficulty: 4.

Prove that the following identity holds for all positive integer \(\displaystyle n\):

\(\displaystyle \sqrt n \le 1+\frac{1}{\sqrt 2} +\ldots + \frac{1}{\sqrt n } <2\sqrt n .\)

Difficulty: 4.

Prove that

\(\displaystyle 1+\frac{1}{2\cdot \sqrt 2} +\ldots + \frac{1}{n\cdot \sqrt n } \le 3- \frac{2}{\sqrt n}. \)

Difficulty: 4.

At least how many steps do you need to move the 64 stories high Hanoi tower?

Towers of Hanoi

Difficulty: 5.

For how many parts the plane is divided by \(\displaystyle n\) lines if no 3 of them are concurrent?

Difficulty: 5.

Prove that finitely many lines or circles divides the plane into domains which can be colored with two colors such that no neighboring domains have the same color.

Difficulty: 5.

Express the following sums in closed forms!

 (1)  \(\displaystyle 1+3+5+7 +\ldots + (2n+1);\)    
 (2)  \(\displaystyle \displaystyle \frac{1}{1\cdot 2\cdot 3} +\ldots +\frac{1}{n\cdot (n+1) \cdot (n+2)} ;\)
 (3)  \(\displaystyle 1\cdot 2 + \ldots +n\cdot (n+1);\)    
 (4)  \(\displaystyle 1\cdot 2\cdot 3+\ldots + n\cdot (n+1)\cdot (n+2).\)

Difficulty: 5.

\(\displaystyle A_1,A_2,\ldots\) are logical statements. What can we say about their truth value if

(a) \(\displaystyle A_1\wedge\forall n\in\N~A_n\Rightarrow A_{n+1}\)?

(b) If \(\displaystyle A_1\wedge\forall n\in\N~A_n\Rightarrow(A_{n+1}\wedge A_{n+2})\)?

(c) If \(\displaystyle A_1\wedge\forall n\in\N~(A_n\vee A_{n+1})\Rightarrow A_{n+2}\)?

(d) If \(\displaystyle \forall n\in N ~ \neg A_n\Rightarrow\exists k\in\{1,2,\ldots,n-1\} ~ \neg A_k\)?

Difficulty: 5.

We cut a corner square of a \(\displaystyle 2^n\) by \(\displaystyle 2^n\) chessboard. Prove that the rest can be covered with disjoint \(\displaystyle L\)-shaped dominoes consisting of 3 squares.

Difficulty: 6.

Show that for all positive integer \(\displaystyle n\geq 6\) a square can be divided into \(\displaystyle n\) squares.

Difficulty: 6. Solution is available for this problem.

We cut two diagonally opposite corner squares of a chessboard. Can we cover the left with \(\displaystyle 1\times 2\) dominoes? And for the \(\displaystyle n\times k\) chessboard?

Difficulty: 7.

Consider the set \(\displaystyle H:=\{2,3,\ldots n+1\}\). Prove that

\(\displaystyle \sum_{\emptyset\neq S\subset H}\prod_{i\in S}\frac1i=n/2. \)

(For example for \(\displaystyle n=3\) we have \(\displaystyle {1\over 2}+\frac13+{1\over 4}+{1\over 2\cdot 3}+{1\over 2\cdot 4}+ {1\over 3\cdot 4}+{1\over 2\cdot 3\cdot 4}={3\over 2}\)).

Difficulty: 7.

Prove that \(\displaystyle \tg1^o\) is irrational!

Difficulty: 7.

For how many parts the space is divided by \(\displaystyle n\) planes if no 4 of them have a common point and no 3 of them have a common line?

Difficulty: 8.