Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.

Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent: (a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$. (b) $(A,+,\cdot)$ is a field.

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$. [i]Daniel Jinga[/i]

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$? [asy]//Choice A size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125; } real g(real x) { return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(A)}$", (-5,4.5)); [/asy] [asy]//Choice B size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2; } real g(real x) { return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(B)}$", (-5,4.5)); [/asy] [asy]//Choice C size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.21875 x^2+0.28125 x+0.5; } real g(real x) { return -0.375 x^2-0.75 x+0.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(C)}$", (-5,4.5)); [/asy] [asy]//Choice D size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875; } real g(real x) { return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625; } real z=3.14; draw(graph(f,-z, z), heavygray); draw(graph(g,-z, z), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(D)}$", (-5,4.5)); [/asy] [asy]//Choice E size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598; } real g(real x) { return -0.166667 x^3+0.125 x^2+0.479167 x-0.375; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(E)}$", (-5,4.5)); [/asy]

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.

$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$

Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder. [i]Proposed by Yahya Motevassel[/i]

Prove there exists a constant $k_0$ such that for any $k\ge k_0$, the equation \[a^{2n}+b^{4n}+2013=ka^nb^{2n}\] has no positive integer solutions $a,b,n$. [i]Proposed by István Pink.[/i]

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $

A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$. [asy] draw((30,0)--(-70,0), Arrow); draw((30,0)--(-20,-40)); draw((-20,-40)--(80,-40), Arrow); draw((0,-60)--(-40,20), dashed, Arrow); draw((0,-60)--(0,15), dashed); draw((0,15)--(40,-65),dashed, Arrow); [/asy]

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

Let \[ f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) \] and \[ g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) \] be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $ 6$. [i]L. Redei[/i]

The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]

Let $k$ be an integer number and $P(X)$ be the polynomial $$P(X) = X^{1997}-X^{1995} +X^2-3kX+3k+1$$ Prove that: a) the polynomial has no integer root; β) the numbers $P(n)$ and $P(n) + 3$ are relatively prime, for every integer $n$.

For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$

The monic polynomial $f$ has rational coefficients and is irreducible over the rational numbers. If $f(\sqrt5 +\sqrt2)= 0$, compute $f(f(\sqrt5 -\sqrt2))$. (A polynomial is [i]monic [/i] if its leading coefficient is $1$. A polynomial is [i]irreducible [/i] over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, $x^2 - 2$ is irreducible, but $x^2 - 1 = (x + 1)(x - 1)$ is not.)

Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?