Found problems: 4275
Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$. If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$?
[i]2016 CCA Math Bonanza Team #6[/i]
For every positive integer $n$, we consider the polynomial of real coefficients, of $2n+1$ terms,
$$P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$$
where all coefficients are real numbers satisfying $100 \le a_i \le 101$ for $0 \le i \le 2n$. Find the smallest possible value of $n$ such that the polynomial can have at least one real root.
Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.
Let $f$ and $g$ be two polynomials such that $f(g(x))=g(f(x))$. If $g(x)$ is linear but not identically equal to $x$, and $f(x)=x^3+60x^2+1000x+c$ for some $c$, find the value of $c$.
[i]Clarification[/i]: $g$ is not constant.
Let $p(x)$ be a polynomial with rational coefficients, of degree at least $2$. Suppose that a sequence $(r_{n})$ of rational numbers satisfies $r_{n}= p(r_{n+1})$ for every $n\geq 1$. Prove that the sequence $(r_{n})$ is periodic.
Let $P(x,y)$ be a polynomial with real coefficients in the variables $x,y$ that is not identically zero. Suppose that $P(\lfloor 2a \rfloor, \lfloor 3a\rfloor) = 0$ for all real numbers $a.$ If $P$ has the minimum possible degree and the coefficient of the monomial $y$ is $4,$ find the coefficient of $x^2y^2$ in $P.$
(The [i]degree[/i] of a monomial $x^my^n$ is $m + n.$ The [i]degree[/i] of a polynomial $P(x,y)$ is then the maximum degree of any of its monomials.)
Let $p(x)$ be an $n$-degree $(n \ge 2)$ polynomial with integer coefficients. If there are infinitely many positive integers $m$, such that $p(m)$ at most $n -1$ different prime factors $f$, prove that $p(x)$ has at most $n-1$ different rational roots .
[color=#f00]a help in translation is welcome[/color]
There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$
Suppose that $P$ is the polynomial of least degree with integer coefficients such that $$P(\sqrt{7} + \sqrt{5}) = 2(\sqrt{7} - \sqrt{5})$$Find $P(2)$.
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have
$$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$
[i]Allen Wang[/i]
Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$?
[i]Adam Ardeishar and Michael Ren[/i]
Let p>3 be a prime and k>0 an integer. Find the multiplicity of X-1 in the factorization of
$ f(X)= X^{p^k-1}+X^{p^k-2}+\cdots+X+1$
modulo p;
in other words, find the unique non-negative integer r such that $ (X - 1)^r $ divides f(X)
\modulo p, but$ (X - 1)^{r+1} $does not divide f(X) \modulo p.
Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
Find all polynomials P(x) satisfying the equation
$(2x-1)P(x) = (x-1)P(2x), \forall x.$
The sum of distinct real roots of the polynomial $x^5+x^4-4x^3-7x^2-7x-2$ is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -2 \qquad\textbf{(E)}\ 7$
Find all pair of constants $(a,b)$ such that there exists real-coefficient polynomial $p(x)$ and $q(x)$ that satisfies the condition below.
[b]Condition[/b]: $\forall x\in \mathbb R,$ $ $ $p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b$
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.
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?
Let $ K$ be a finite field of $ p$ elements, where $ p$ is a prime. For every polynomial
$ f(x)\equal{}\sum_{i\equal{}0}^na_ix^i$ ($ \in K[x]$)
put
$ \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}$.
Prove that for any pair of polynomials $ f(x),g(x)\in K[x]$, $ \overline{f(x)}|\overline{g(x)}$ if and only if $ f(x)|g(x)$.
Prove that there is no polynomial $P \in \mathbb C[x]$ such that set $\left \{ P(z) \; | \; \left | z \right | =1 \right \}$ in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon.
(30 points)
Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.
Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$.
[i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]