Let both $A$ and $B$ be two-digit numbers, and their difference is $14$. If the last two digits of $A^2$ and $B^2$ are the same, what are all the possible values of $A$ and $B$.
There is a sequence with $a(2) = 0$, $a(3) = 1$ and $a(n) = a(\lfloor{\frac{n}{2}}\rfloor)+a(\lceil{\frac{n}{2}}\rceil)$ for $n\ge 4$. Find $a(2014)$.
Evaluate $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{1368}+\sqrt{1369}}$.
How many integer pairs $(a,b)$ with $1 < a, b\le 2015$ are there such that $log_a b$ is an integer?
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 is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?
What are all values of $x$ for which $log_x\sqrt{x+12}>1$?
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 $$1+\frac{1}{2+\frac{1}{3+a}}$$
Imagine there is an infinitive grid. Each grid is a square with side length of 1. Find the ratio of the number of points, number of unit squares and the number of sides of these unit squares.
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible?
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?
Let $x$ be a positive number. Denote by $[x]$ the integer part of $x$ and by $\{x\}$ the decimal part of $x$. Find the sum of all positive numbers satisfying $5\{x\} + 0.2[x] = 25$.
Find the largest integer not exceeding $1 + \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}} + \cdots + + \frac{1}{\sqrt{10000}}$
For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$?
The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by \[\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor\]where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )$?
The graphs of $y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,$ and $y=\log_x \dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs?
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$?
Let $x=-2016$. What is the value of $\bigg|$ $|x|-x|-|x|$ $\bigg|$ $-x$?
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$?
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?
Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.