Found problems: 3597
A positive integer $n$ is given. Consider all polynomials $P(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_0$, whose coefficients are nonnegative integers, not exceeding $100$. Call $P$ [i]reducible[/i] if it can be factored into two non-constant polynomials with nonnegative integer coeffiecients, and [i]irreducible[/i] otherwise. Prove that the number of [i]irreducible[/i] polynomials is at least twice as big as the number of [i]reducible[/i] polynomials.
[i]D. Zmiaikou[/i]
Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
Let $a$ be the root of the equation $x^3-x-1=0$. Find an equation of the third degree with integer coefficients whose root is $a^3$.
Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients.
[i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]
Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$?
$ \textbf{(A)}\ 19\qquad
\textbf{(B)}\ 22\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 27\qquad
\textbf{(E)}\ 32$
Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.
Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]
The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of
\[ x^2 \plus{} px \plus{} q \equal{} 0
\]
to be $ r^2$ and $ s^2$, $ p$ must equal:
$ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad
\textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad
\textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\
\textbf{(D)}\ B^2 \minus{} 2C\qquad
\textbf{(E)}\ 2C \minus{} B^2$
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.
[i]Proposed by Carl Schildkraut[/i]
Let $ a\ge 3 $ and a polynom $ P. $ Show that:
$$ \max_{1\le k\le \text{grad} P} \left| a^{k-1}-P(k-1) \right| \ge 1 $$
$(POL 3)$ 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 $m.$
Find all real polynomials $P(x)$ such that for every four distinct natural numbers $a, b, c, d$ such that $a^2 + b^2 + c^2 = 2d^2$ with $gcd(a, b, c, d) = 1$ the following equality holds:
$$2(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2$$
.
Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies
\[
\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},
\]
compute the ordered triple $(a, b, c)$.
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:\\
\[f(a)=bc , f(b)=ac, f(c)=ab\]
Dertermine $f(a+b+c)$ in terms of $a,b,c$.
(a) Let $a_0, a_1,a_2$ be real numbers and consider the polynomial $P(x) = a_0 + a_1x + a_2x^2$ .
Assume that $P(-1), P(0)$ and $P(1)$ are integers.
Prove that $P(n)$ is an integer for all integers $n$.
(b) Let $a_0,a_1, a_2, a_3$ be real numbers and consider the polynomial $Q(x) = a0 + a_1x + a_2x^2 + a_3x^3 $.
Assume that there exists an integer $i$ such that $Q(i),Q(i+1),Q(i+2)$ and $Q(i+3)$ are integers.
Prove that $Q(n)$ is an integer for all integers $n$.
Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation
\[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\]
\[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\]
\[+(x - b)(x - c)(x - d)(x - e) = 0\]
has four distinct real solutions.
If $P(x)=x^n+a_1x^{n-1}+...+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\]
(We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.)
[i]Proposed by the4seasons.[/i]