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 $\frac{m}{n}$?
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 $k$
What is the $22^{nd}$ positive integer $n$ such that $22^n$ ends in a $2$?
Find the sum of all positive integers $n$ such that the least common multiple of $2n$ and $n^2$ equals $(14n - 24)$?
In the diagram, AB is the diameter of a semicircle, D is the midpoint of the semicircle, angle $BAC$ is a right angle, $AC=AB$, and $E$ is the intersection of $AB$ and $CD$. Find the ratio between the areas of the two shaded regions.
Find the largest 7-digit integer such that all its 3-digit subpart is either a multiple of 11 or multiple of 13.
Find a $4$-digit square number $x$ such that if every digit of $x$ is increased by 1, the new number is still a perfect square.
What is Heron's formula to calculate a triangle's area given the lengths of three sides?
If the first $25$ positive integers are multiplied together, in how many zeros does the product terminate?
What is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?
Of the pairs of positive integers $(x, y)$ that satisfies $3x+7y=188$, which ordered pair has the least positive difference $x-y$?
What is the smallest positive integer greater than $5$ which leaves a remainder of $5$ when divided by each of $6$, $7$, $8$, and $9$?
What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?
How many minutes past $4$ o'clock are the hands of a standard $12$-hour clock first perpendicular to each other?
Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?
Determine the units digit of the sum $0!+1!+2!+\cdots+n!+\cdots+20!$?
What real value of $x$ satisfies $\sqrt{5x} - \sqrt{2x} = 5-2$?
Let point $A$ and $B$ represent real numbers $a$ and $b$, respectively. If $A$ and $B$ lay on different sides of the origin $O$, and $|a - b| = 2016$, $AO = 2BO$, what is the value of $a+b$?
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 $$1+\frac{1}{2+\frac{1}{3+a}}$$
How many ordered pairs of integers $(x,y)$ are there such that $x^2 + 2xy+3y^2=34$?
Show that $$\binom{n}{k} = \frac{n}{k}\binom{n-1}{k-1}$$
Let positive integers $m\le k \le n$. Show that $$\binom{n}{k}\binom{k}{m} =\binom{n}{m}\binom{n-m}{k-m} =\binom{n}{k-m}\binom{n-k+m}{m}$$