Found problems: 27
Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.
Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009 \\
& x^3 + 5x^2-49x-245 \\
& x^3 + 39x^2 - 117x - 1435\end{align*}
Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009 \\
& x^3 + 5x^2-49x-245 \\
& x^3 + 39x^2 - 117x - 1435\end{align*}
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $
Let rational numbers $a, b, c$ satisfy the conditions $a + b + c = a^2 + b^2 + c^2 \in Z$.
Prove that there exist two relative prime numbers $m, n$ such that $abc =\frac{m^2}{n^3}$ .
Find the smallest positive integer $ n$ with the property that the polynomial $ x^4 \minus{} nx \plus{} 63$ can be written as a product of two nonconstant polynomials with integer coefficients.
Find all integers $ k$ such that for infinitely many integers $ n \ge 3$ the polynomial
\[ P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995\]
can be reduced into two polynomials with integer coefficients.
Solve the following system for real $x,y,z$
\[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]
For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$?
$\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$
Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.
Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$
Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?
The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$.
$\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$
Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.
Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$.
Find $24\cot^2{x}$.
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.
Is rational or irrational,the number
$$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$
Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers
$c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$
are all integers.
Solve the following system of equations in rational numbers:
\[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$
Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$