Found problems: 3597
The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is
$\text{(A)} \ 5 \qquad \text{(B)} \ 7 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 17 \qquad \text{(E)} \ 72$
Find all polynomials $p(x)$ with real coefficients that satisfy $$4p(x^2) = 4(p(x))^2 + 4p(x)- 1$$
Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers?
[i]Proposed by Morteza Saghafian[/i]
Which of the following four statements are true and which are false?
a) If a polygon inscribed in a circle is equilateral, then it is also equiangular.
b) If a polygon inscribed in a circle is equiangular, then it is also equilateral.
c) If a polygon circumscribed to a circle is equilateral, then it is also equiangular.
d) If a polygon circumscribed to a circle is equiangular, then it is also equilateral.
For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.
A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.
Does every positive polynomial in two real variables attain its lower bound in the plane?
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
Let $d\geq 13$ be an integer, and let $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x+a_0$ be a polynomial of degree $d$ with complex coefficients such that $a_n = a_{d-n}$ for all $n\in\{0,1,\dots,d\}$. Prove that if $P$ has no double roots, then $P$ has two distinct roots $z_1$ and $z_2$ such that $|z_1-z_2|<1$.
Let $P=(x^4-40x^2+144)(x^3-16x)$.
$a)$ Factor $P$ as a product of irreducible polynomials.
$b)$ We write down the values of $P(10)$ and $P(91)$. What is the greatest common divisor of the two numbers?
Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that
$$P(x)=\sum_{}{x^{n_i}}+1$$
Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.
Suppose that $n$ is a natural number and $n$ is not divisible by $3$. Prove that
$(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1$
has at least $2d(n)$ distinct prime factors where $d(n)$ is the number of positive divisors of $n$.
[i]proposed by Mahyar Sefidgaran[/i]
Call a non-constant polynomial [i]real[/i] if all its coecients are real. Let $P$ and $Q$ be polynomials with complex coefficients such that the composition $P \circ Q$ is real. Show that if the leading coefficient of $Q$ and its constant term are both real, then $P$ and $Q$ are real.
Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.
Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$. Determine the maximum possible value of degree of $T(x)$
Let $k$, $\ell $ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^\circ$ and let its sidelengths be $AB=k$, $BC=\ell$ and $CD=m$. Let $f(k,\ell,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^\circ$.
Prove that by fixing $\ell$ and $m$, there exists polynomial $g_{\ell,m}$ such that $f(k,\ell,m)=g_{\ell,m}(k)$ for every positive integer $k$, and find the degree of $g_{\ell,m}$ in terms of $\ell$ and $m$.
[i]Submitted by Zoltán Gyenes, Budapest[/i]
Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.
For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that:
$P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$.
Prove that $2P(2n + 1) - P(2n + 2) = 1$.
P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)
Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$.
for all $n \geq 0$.
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n.
We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.
Let $ P[X,Y] $ be a polynomial of degree at most $ 2 .$ If $ A,B,C,A',B',C' $ are distinct roots of $ P $ such that $ A,B,C $ are not collinear and $ A',B',C' $ lie on the lines $ BC,CA, $ respectively, $ AB, $ in the planar representation of these points, show that $ P=0. $