Found problems: 4275
Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$
Let $ f(x) \equal{} \prod^n_{k\equal{}1} (x \minus{} a_k) \minus{} 2,$ where $ n \geq 3$ and $ a_1, a_2, \ldots,$ an are distinct integers. Suppose that $ f(x) \equal{} g(x)h(x),$ where $ g(x), h(x)$ are both nonconstant polynomials with integer coefficients. Prove that $ n \equal{} 3.$
The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.
The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.
$f(x) = x^3-ax+1$ , $a \in R$ has three different zeros in $R$. Prove that for the zero $x_o$ with the smallest absolute value holds: $\frac{1}{a}< x_0 < \frac{2}{a}$
Let $n\ge 2$ be an integer. We call an ordered $n$-tuple of integers primitive if the greatest common divisor of its components is $1$. Prove that for every finite set $H$ of primitive $n$-tuples, there exists a non-constant homogenous polynomial $f(x_1,x_2,\ldots,x_n)$ with integer coefficients whose value is $1$ at every $n$-tuple in $H$.
[i]Based on the sixth problem of the 58th IMO, Brazil[/i]
$p$ is a prime. Let $K_p$ be the set of all polynomials with coefficients from the set $\{0,1,\dots ,p-1\}$ and degree less than $p$. Assume that for all pairs of polynomials $P,Q\in K_p$ such that $P(Q(n))\equiv n\pmod p$ for all integers $n$, the degrees of $P$ and $Q$ are equal. Determine all primes $p$ with this condition.
Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$.
[hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying the equality $P(Q(x))=P(x)Q(x)-P(x)$.
[i](I. Voronovich)[/i]
Let $P$ be a polynomial. Suppose that there exists a rational $q$ such that $P(m)=q^n$ for infinitely many integers $(m,n)$. Prove that $P(x)=c\cdot Q(x)^k$ for some integer constants $c$ and $k$ and irreducible polynomial $Q$ with rational coefficients.
(Here, a polynomial is $\textit{irreducible}$ if it can't be factored into the product of non-constant polynomials with rational coefficients.)
[i]Jason Lee[/i]
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold:
(1) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$
(2) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$
(3) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$
Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that \[ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.\]
Let $n$ be a positive integer, $n \geq 2$, and consider the polynomial equation
\[x^n - x^{n-2} - x + 2 = 0.\]
For each $n,$ determine all complex numbers $x$ that satisfy the equation and have modulus $|x| = 1.$
Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$.
Determine the number of quadratic polynomials $P(x) = p_1x^2 + p_2x - p_3$, where $p_1$, $p_2$, $p_3$ are not necessarily distinct (positive) prime numbers less than $50$, whose roots are distinct rational numbers.
Determine the number of real roots of the equation
${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$
Find all polynomials $ P(x,y)$ such that for all reals $ x$ and $y$,
\[P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).\]
We say that a prime $p$ is $\textit{philé}$ if there is a polynomial $P$ of non-negative integer coefficients smaller than $p$ and with degree $3$, that is, $P(x) = ax^3 + bx^2 + cx + d$ where $a, b, c, d < p$, such that $$\{P(n) | 1 \leq n \leq p\}$$ is a complete residue system modulo $p$. Find all $\textit{philé}$ primes.
Note: A set $A$ is a complete residue system modulo $p$ if for every integer $k$, with $0 \leq k \leq p - 1$, there exists an element $a \in A$ such that $$p | a-k.$$
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
Do there exist two polynomials $P$ and $Q$ with integer coefficient such that
i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$,
ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.
Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$.
Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$.
Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).
Let us consider every third degree polynomial $P(x)$ with coefficients as nonnegative positive integers such that $P(1)=20$. Among them determine polynomial for which is:
$a)$ Minimal value of $P(4)$
$b)$ Maximal value of $P(3)/P(2)$
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$
Assume that there exists a prime number $p$ satisfying these conditions:
[i]i)[/i] $p|a_i$ for all $0\le i<n,$
[i]ii)[/i] $p\nmid a_n,$
[i]iii)[/i] $p^2\nmid a_0.$
Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
For each positive integer $n$, determine the smallest possible value of the polynomial
$$
W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx.
$$