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 The incircle of triangle $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ at $D, E, F$ respectively. $X$ is a point inside triangle of $\triangle ABC$ such that the incircle of triangle $\triangle XBC$ touches $BC$ at $D$, and touches $CX$ and $XB$ at $Y$ and $Z$ respectively. Show that $E, F, Z, Y$ are concyclic.

%V0 An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$), $B_1$ and $B_2$ on the side $AC$ (with $B_2$ between $B_1$ and $A$), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$) so that $$\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.$$ The lines $AA_1,BB_1,$ and $CC_1$ bound a triangle, and the lines $AA_2,BB_2,$ and $CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

%V0 Determine all integers $n > 3$ for which there exist $n$ points $A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $r_{1},\cdots ,r_{n}$ such that for $1\leq i < j < k\leq n$, the area of $\triangle A_{i}A_{j}A_{k}$ is $r_{i} + r_{j} + r_{k}$.

%V0 For an integer $x \geq 1,$ let $p(x)$ be the least prime that does not divide $x,$ and define $q(x)$ to be the product of all primes less than $p(x).$ In particular, $p(1) = 2.$ For $x$ having $p(x) = 2,$ define $q(x) = 1.$ Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and $$x_{n+1} = \frac{x_n p(x_n)}{q(x_n)}$$ for $n \geq 0.$ Find all $n$ such that $x^n = 1995.$

%V0 Suppose that $x_1, x_2, x_3, \ldots$ are positive real numbers for which $$x^n_n = \sum^{n-1}_{j=0} x^j_n$$ for $n = 1, 2, 3, \ldots$ Prove that $\forall n,$ $$2 - \frac{1}{2^{n-1}} \leq x_n < 2 - \frac{1}{2^n}.$$