Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.

Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

Let \( A \) and \( B \) be two square matrices with complex entries such that \( A + B = AB \), \( A = A^* \), and \( A \) has all distinct eigenvalues. Prove that there exists a polynomial \( P \) with complex coefficients such that \( P(A) = B \).

Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$. $\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$

Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials $P(x)$ and $Q(x)$ commute, if $P(Q(x))=Q(P(x))$ (i.e. we obtain the same polynomial, having collected the similar terms). a) For every a find all $Q$ such that the $Q$ degree is not greater than three, and $Q$ commutes with $(x^2 - a)$. b) Let $P$ be a square polynomial, and $k$ is a natural number. Prove that there is not more than one commuting with $P$ $k$-degree polynomial. c) Find the $4$-degree and $8$-degree polynomials commuting with the given square polynomial $P$. d) $R$ and $Q$ commute with the same square polynomial $P$. Prove that $Q$ and $R$ commute. e) Prove that there exists a sequence $P_2, P_3, ... , P_n, ...$ ($P_k$ is $k$-degree polynomial), such that $P_2(x) = x^2 - 2$, and all the polynomials in this infinite sequence pairwise commute.

For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied: \[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

A cubic polynomial $f$ satisfies $f(0)=0, f(1)=1, f(2)=2, f(3)=4$. What is $f(5)$?

$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] (a) Prove that there are a finite number of natural numbers in range of $f$. (b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions. (c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers. Time allowed for this problem was 105 minutes.

Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

A given polynomial $P(t) = t^3 + at^2 + bt + c$ has $3$ distinct real roots. If the equation $(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0$ has no real roots, prove that $P(2013) >\frac{1}{64}$

Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

[i]The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).[/i] Let $p(x)$ be a polynomial of degree $d$ with coefficients belonging to the set of rational numbers $\mathbb{Q}$. Suppose that, for each $1 \le k \le d-1$, $p(x)$ and its $k$th derivative $p^{(k)}(x)$ have a common root in $\mathbb{Q}$; that is, there exists $r_k \in \mathbb{Q}$ such that $p(r_k) = p^{(k)}(r_k) = 0$. (a) Prove that if $d$ is prime then there exist constants $a, b, c \in \mathbb{Q}$ such that \[ p(x) = c(ax + b)^d. \] (b) For which integers $d \ge 2$ does the conclusion of part (a) hold?

Find $a\in Z$ such that the equation $2x^2+2ax+a-1=0$ has integer solutions, which should be found.

Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.

Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$

Find all values of parameters $a,b$ for which the polynomial $$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$.

Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.

Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$? $ \textbf{(A)}\ 840 \qquad\textbf{(B)}\ 816 \qquad\textbf{(C)}\ 969 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $