Found problems: 4275
Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
[asy]
real r=(3+sqrt(5))/2;
real s=sqrt(r);
real Brad=r;
real brad=1;
real Fht = 2*s;
import graph3;
import solids;
currentprojection=orthographic(1,0,.2);
currentlight=(10,10,5);
revolution sph=sphere((0,0,Fht/2),Fht/2);
//draw(surface(sph),green+white+opacity(0.5));
//triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));}
triple f(pair t) {
triple v0 = Brad*(cos(t.x),sin(t.x),0);
triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht);
return (v0 + t.y*(v1-v0));
}
triple g(pair t) {
return (t.y*cos(t.x),t.y*sin(t.x),0);
}
surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2);
surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2);
surface base = surface(g,(0,0),(2pi,Brad),80,2);
draw(sback,rgb(0,1,0));
draw(sfront,rgb(.3,1,.3));
draw(base,rgb(.4,1,.4));
draw(surface(sph),rgb(.3,1,.3));
[/asy]
$ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $
Let $p(x)$ be a polynomial with real coefficients and $n$ be a positive integer. Prove that there exists a non-zero polynomial $q(x)$ with real coefficients such that the polynomial $p(x)\cdot q(x)$ has non-zero coefficients only by the powers which are multiples of $n$.
Find all polynomials $ P (x) $ with real coefficients that satisfy the identity $ P (x ^ 2) = P (x) P (x + 1) $.
Let $a$, $b$, $c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$.
The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2$, $ca-b^2$, $ab-c^2$.
What is the sum of the coefficients of $Q$?
[i]Proposed by Evan Chen[/i]
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
Prove that every polynomial is a difference of two increasing polynomials.
Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
Let $P_1$, $P_2$ and $P_3$ be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial $P_1 + P_2 + P_3$ has also real roots.
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?
[i]A. Khrabrov[/i]
Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$
Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$,
$P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds:
$$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$
Let $P(x)$ be a polynomial with real coefficients. Prove that not all roots of $x^3P(x)+1$ are real.
[b]7.[/b] Prove that for any odd prime number $p$, the polynomial
$2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$
is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b]
*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.
Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$
$\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$
The equation $P(x) = 0$, where $P(x) = x^2+bx+c$, has a single root, and the equation $P(P(P(x))) = 0$ has exactly three different roots. Solve the equation $P(P(P(x))) = 0.$
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
Determine all monic polynomials $P(x)$ of degree $3$ with coefficients integers, which are divisible by $x-1$, when divided by $ x-5$ leave the same remainder as when divided by$ x+5$ and have a root between $2$ and $3$.