Found problems: 4275
$2.$ 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 $k ?$
Find $A=x^5+y^5+z^5$ if $x+y+z=1$, $x^2+y^2+z^2=2$ and $x^3+y^3+z^3=3$.
Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the fifth powers of the roots of $P$.
Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$
for all reals $x,y$.
Let $k$ be a positive integer larger than $1$. Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:
[b](a)[/b] any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;
[b](b)[/b] any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
Find all nonzero polynomials $P(x)$ with integers coefficients that satisfy the following property: whenever $a$ and $b$ are relatively prime integers, then $P(a)$ and $P(b)$ are relatively prime as well. Prove that your answer is correct. (Two integers are [b]relatively prime[/b] if they have no common prime factors. For example, $-70$ and $99$ are relatively prime, while $-70$ and $15$ are not relatively prime.)
Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
Is it true that for any polynomial $P(x)$ with real coefficients of degree $2023$, there exists a natural number $n$ such that the equation $P(x) = n^{-100}$ has no rational root?
The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
What is the sum of the reciprocals of the roots of the equation
\[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0?
\]
$ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$
Let $P,Q,R$ be the polynomials with real or complex coefficients such that at least one of them is not constant. If $P^n+Q^n+R^n = 0$, prove that $n < 3.$
Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$.
[i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.
Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$
Let $a$ and $b$ be real numbers, and the polynomial $P(x) =ax + b$ such that $P(2)- P(1)= 3$:
Compute the value of $P(5)- P(0)$.
[b]A.[/b] $11$ [b]B.[/b] $13$ [b]C.[/b] $15$ [b]D.[/b] $17$ [b]E.[/b] $19$
Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that
$$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$
for all integer polynomials $ p. $