Školjka

%V0 Find all of the positive real numbers like $x,y,z,$ such that : 1.) $x + y + z = a + b + c$ 2.) $4xyz = a^2x + b^2y + c^2z + abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

%V0 Let $k$ be a positive integer. Show that there are infinitely many perfect squares of the form $n \cdot 2^k - 7$ where $n$ is a positive integer.

%V0 Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$.

%V0 Consider two monotonically decreasing sequences $\left( a_k\right)$ and $\left( b_k\right)$, where $k \geq 1$, and $a_k$ and $b_k$ are positive real numbers for every k. Now, define the sequences $c_k = \min \left( a_k, b_k \right)$; $A_k = a_1 + a_2 + ... + a_k$; $B_k = b_1 + b_2 + ... + b_k$; $C_k = c_1 + c_2 + ... + c_k$ for all natural numbers k. (a) Do there exist two monotonically decreasing sequences $\left( a_k\right)$ and $\left( b_k\right)$ of positive real numbers such that the sequences $\left( A_k\right)$ and $\left( B_k\right)$ are not bounded, while the sequence $\left( C_k\right)$ is bounded? (b) Does the answer to problem (a) change if we stipulate that the sequence $\left( b_k\right)$ must be $\displaystyle b_k = \frac {1}{k}$ for all k ?

%V0 Za svaki prirodan broj $n$ određeni su cijeli brojevi $a_n$ i $b_n$ tako da je $$ (1+\sqrt{2})^{2n+1}=a_n+b_n \sqrt{2}.$$ a) Dokažite da su $a_n$ i $b_n$ neparni za svaki $n$. b) Dokažite da je $b_n$ hipotenuza pravokutnog trokuta čije su katete $$ \frac{a_n+(-1)^n}{2}, \ \frac{a_n-(-1)^n}{2}. $$

%V0 Three strictly increasing sequences $$a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots$$ of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold: (a) $c_{a_n}=b_n+1$; (b) $a_{n+1}>b_n$; (c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even. Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.