Let $x_1$ and $x_2$ be two real roots of $m^2x^2 +2(3-m)x+1=0$. If $m=\frac{1}{x_1}+\frac{1}{x_2}$, what is the value of $m$?
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2(k+1)x+k^2 + 2 = 0$. If $(x_1+1)(x_2+1) =8$, find the value of $k$
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2mx + (m^2+2m+3)=0$. Find the minimal value of $x_1^2 + x_2^2$.
If for any integer $k\ne 27$ and $\big(a-k^{2015}\big)$ is divisible by $(27-k)$, what is the last two digits of $a$?
Let $P(x)$ be a polynomial with integer coefficients. Show that $P(7)=5$ and $P(15)=9$ cannot hold simultaneously.
If $a+b=\sqrt{5}$, compute $\frac{a^2 -a^2b^2 + b^2 +2ab}{a+ab+b}$.
What value of $a$ satisfies $27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$?
What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?
If $a+b=\sqrt{5}$, compute the value of $\frac{a^2 - a^2b^2 + b^2 +2ab}{a+ab+b}+ab$.
Distinct real numbers $a$, $b$ and $c$ satisfy $a+\frac{1}{b}=b+\frac{1}{c} = c+\frac{1}{a}=t$. Find the value of $t$.
If real numbers $a$, $b$ and $c$ satisfy $abc=-1$, $a+b+c=4$, $\frac{a}{a^2-3a-1}+\frac{b}{b^2-3b-1}+\frac{c}{c^2-3c-1}=\frac{4}{9}$, what is the value of $a^2+b^2+c^2$?
Show that $x^n + 5x^{n-1} + 3 = 0$ cannot be factorized into two non-constant polynomials with integer coefficients.
If the $5^{th}$, $6^{th}$ and $7^{th}$ coefficients in the expansion of $(x^{-\frac{4}{3}}+x)^n$ form an arithmetic sequence, find the constant term in the expanded form.
A positive integer is written on each face of a cube. Then for each vertex of the cube, the product of the numbers on the three faces associated with this vertex is calculated. If the sum of these eight products equals 2015, find the sum of all the numbers on the 6 faces.
Let $f(x)= \sqrt{ax^2+bx}$. For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?
Compute the sum of all the roots of $(2x+3)(x-4)+(2x+3)(x-6)=0$
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is
Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
Find the largest positive integer $n$ such that $(3^{1024} - 1)$ is divisible by $2^n$.
Find a polynomial with integral coefficients whose zeros include $\sqrt{2}+\sqrt{5}$.
Let $p(x)$ be a polynomial with integer coefficients. Assume that $p(a) = p(b) = p(c) = -1$, where $a, b, c$ are three different integers. Prove that $p(x)$ has no integral zeros.
Prove that the sum $$\sqrt{1001^2 + 1} + \sqrt{1002^2 + 1} + \cdots + \sqrt{2000^2 + 1}$$ is irrational.
If $P(x)$ denotes a polynomial of degree $n$ such that $P(k) = k/(k +1)$ for $k = 0, 1, 2, \dots n$, determine $P(n + 1)$.
The product of two of the four zeros of the quartic equation $$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$ is $-32$. Find $k$.