If one root of $x^2 + \sqrt{2}x + a = 0$ is $1-\sqrt{2}$, find the other root as well as the value of $a$.
Consider the equation $x^2 +(m-2)x + \frac{1}{2}m-3=0$.
(1) Show that this equation always have two distinct real roots
(2) Let $x_1$ and $x_2$ be its roots. If $x_1+x_2=m+1$, what is the value of $m$?
If $n>0$ and $x^2 -(m-2n)x + \frac{1}{4}mn=0$ has two equal positive real roots, what is the value of $\frac{m}{n}$?
If real number $m$ and $n$ satisfy $mn\ne 1$ and $19m^2+99m+1=0$ and $19+99n+n^2=0$, what is the value of $\frac{mn+4m+1}{n}$?
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 $x_1^2 + x_2^2$.
How many integer pairs $(a,b)$ with $1 < a, b\le 2015$ are there such that $log_a b$ is an integer?
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n =\frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
Suppose that $(u_n)$ is a sequence of real numbers satisfying $u_{n+2}=2u_{n+1}+u_n$, and that $u_3=9$ and $u_6=128$. What is $u_{2015}$?
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 $\frac{a^2 -a^2b^2 + b^2 +2ab}{a+ab+b}$.
Let $m$ be an odd positive integer, and not a multiple of 3. Show that the integer part of $4^m - (2+\sqrt{2})^m$ is a multiple of 112.
How many digits are there if the numbers $2^{2015}$ and $5^{2015}$ are written one after another?
What value of $a$ satisfies $27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$?
What is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?
What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?
What are all values of $x$ for which $log_x\sqrt{x+12}>1$?
If $a+b=\sqrt{5}$, compute the value of $\frac{a^2 - a^2b^2 + b^2 +2ab}{a+ab+b}+ab$.
How many solutions does the following system have?
$$
\left\{
\begin{array}{ll}
\lfloor x \rfloor + 2y &= 1\\
\lfloor y \rfloor + x &=2
\end{array}
\right.
$$
Where $\lfloor x \rfloor$ and $\lfloor y \rfloor$ denote the largest integers not exceeding $x$ and $y$, respectively.
Let $a=-2+\sqrt{2}$. Compute $$1+\frac{1}{2+\frac{1}{3+a}}$$
Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.
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 $a^2+b^2+c^2$?