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 $a$?
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?
(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$
Solve $4x^2+27x-9\equiv 0\pmod{15}$
Solve $5x^3 -3x^2 +3x-1\equiv 0\pmod{11}$
Solve $3x^{15}-x^{13}-x^{12} -x^{11} -3x^5 +6x^3 -2x^2 +2x-1\equiv 0 \pmod{11}$
Solve $14x\equiv 30 \pmod{21}$
Solve $17x\equiv 229\pmod{1540}$.
Solve $$\left\{ \begin{array}{rcl} x &\equiv 2 &\pmod{3}\\ x &\equiv 2 &\pmod{5}\\ x &\equiv -3 &\pmod{7}\\x &\equiv -2 &\pmod{13} \end{array}\right.$$
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?
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 $\sqrt{5x} - \sqrt{2x} = 5-2$?
Solve $$\left\{ \begin{array}{rcl} 4x & \equiv 14 &\pmod{15}\\ 9x & \equiv 11 &\pmod{20}\\ \end{array}\right.$$
Show that the sum of all the numbers of the form $\frac{1}{mn}$ is not an integer, where $m$ and $n$ are integers, and $1\le m \le n \le 2017$.
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.
Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.
How many ordered pairs of integers $(x,y)$ are there such that $x^2 + 2xy+3y^2=34$?
If real numbers $a$ and $b$ satisfy $a^2 + b^2=1$, find the minimal value of $a^4 + ab+b^4$.
Show that $x^n + 5x^{n-1} + 3 = 0$ cannot be factorized into two non-constant polynomials with integer coefficients.