Found problems: 3597
Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$.
Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$.
[i](Paolo Leonetti)[/i]
Let $$p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}$$ Show that it is non-negative for all real $x$ and all positive integers $k$.
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.
Let $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form
\[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$.
[i]Valentin Vornicu[/i]
Consider the equation $n^2+(n+1)^4=5(n+2)^3$
(a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation.
(b) Does the equation have a solution in positive integers?
Let be two (not necessarily distinct) roots of two rational polynoms (respectively) that are irreducible over the rationals. Prove that these polynoms have the same degree if the sum of those two roots is rational.
[i]Bogdan Enescu[/i]
$P(x)$ is a nonconstant polynomial with real coefficients and its all roots are real numbers. If there exist a $Q(x)$ polynomial with real coefficients that holds the equality for all $x$ real numbers
$(P(x))^{2}=P(Q(x))$,
then prove that all the roots of $P(x)$ are same.
Determine all pairs of real numbers $ a, b $ for which the polynomials $ x^4 + 2ax^2 + 4bx + a^2 $ and $ x^3 + ax - b $ have two different common real roots.
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.
Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.
Determine the smallest possible degree of $f$.
(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)
[i]Ankan Bhattacharya[/i]
For what is the smallest natural number $n$ there is a polynomial $P(x)$ with integer coefficients, having $m$ different integer roots, and at the same time the equation $P(x) = n$ has at least one integer solution if:
a) $m = 5$,
b) $ m = 6$?
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle
Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.
Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.
Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.
Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$ , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity.
[i]Proposed by Mr.Etesami[/i]
Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that
(i) $a_n>0, \; \forall n \in \mathbb{N}$.
(ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$.
Prove that $|p-1| \ge 2$.
Solve the system equations
\[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]
Let $P(x)$ be a polynomial of degree $n \ge 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.
Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.