Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$

[b]a)[/b] Let $ a,b,c $ be three complex numbers. Prove that the element $ \begin{pmatrix} a & a-b & a-b \\ 0 & b & b-c \\ 0 & 0 & c \end{pmatrix} $ has finite order in the multiplicative group of $ 3\times 3 $ complex matrices if and only if $ a,b,c $ have finite orders in the multiplicative group of complex numbers. [b]b)[/b] Prove that a $ 3\times 3 $ real matrix $ M $ has positive determinant if there exists a real number $ \lambda\in\left( 0,\sqrt[3]{4} \right) $ such that $ A^3=\lambda A+I. $ [i]Cristinel Mortici[/i]

Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i]. (a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent. (b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.

[b]2.[/b] Construct a sequence $(a_n)_{n=1}^{\infty}$ of complex numbers such that, for every $l>0$, the series $\sum_{n=1}^{\infty} \mid a_n \mid ^{l}$ be divergent, but for almost all $\theta$ in $(0,2\pi)$, $\prod_{n=1}^{\infty} (1+a_n e^{i\theta})$ be convergent. [b](S. 11)[/b]

Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.

Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.

What is the largest amount of complex $ z $ solutions a system can have? $ | z-1 || z + 1 | = 1 $ $ Im (z) = b? $ (where $ b $ is a real constant)

Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.

Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of $$\frac{KM + LN}{AC + BD}$$ .

Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$). We define function $f:S\rightarrow S$ as follow: $\forall z\in S$, $ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$ $f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$ We call $c$ an $n$-[i]period-point[/i] of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy: $f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$. Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-[i]period-point[/i] of $f$.

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following: [list] [*] For marked points $X,Y$, mark the reflection of $X$ across $Y$. [*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$. [*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$. [/list] No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves, a) The circumcenter of $\triangle ABC$. b) The incenter of $\triangle ABC$. Proposed by [i]Deniz Can Karaçelebi[/i]

[b]2.[/b] Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers. [b](F. 2)[/b]

Let $ z_1,z_2,z_3\in\mathbb{C} , $ different two by two, having the same modulus $ \rho . $ Show that: $$ \frac{1}{\left| z_1-z_2\right|\cdot \left| z_1-z_3\right|} +\frac{1}{\left| z_2-z_1\right|\cdot \left| z_2-z_3\right|} +\frac{1}{\left| z_3-z_1\right|\cdot \left| z_3-z_2\right|}\ge\frac{1}{\rho^2} . $$

What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?

Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$ If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold \[z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0\] prove that [b]a)[/b] Triangle $A_1A_3A_5$ is equilateral [b]b)[/b] \[|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.\]

Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.

Consider the second degree polynomial $x^2+ax+b$ with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, $a^2-4b$ be greater than or equal to zero. Note that the discriminant is also a polynomial with variables $a$ and $b$. Prove that the same story is not true for polynomials of degree $4$: Prove that there does not exist a $4$ variable polynomial $P(a,b,c,d)$ such that: The fourth degree polynomial $x^4+ax^3+bx^2+cx+d$ can be written as the product of four $1$st degree polynomials if and only if $P(a,b,c,d)\ge 0$. (All the coefficients are real numbers.) [i]Proposed by Sahand Seifnashri[/i]

Determine all roots, real or complex, of the system of simultaneous equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}

Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.