What is the smallest positive integer $n$ such that $20\equiv n^{15} \pmod{29}$?
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3\le p<100$, and $1\le a < p$ such that the sum $a+a^2+a^3+ \cdots + a^{(p-2)!}$ is not divisible by $p$?
Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.)
- Find, with proof, the maximum possible value of $n$.
- For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard.
For every integer $n$, let $m$ denote the integer made up of the last four digit of $n^{2015}$. Consider all positive integer $n < 10000$, let $A$ be the number of cases when $n > m$, and $B$ be the number of cases when $n < m$. Compute $A-B$.
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.
Let $\frac{p}{q}=1+ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100000}$ where $p$ and $p$ are both positive integers and do not have common divisor greater than 1. How many ending zeros does $q$ have?
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.
Let $m$ be a positive odd integer, $m\ge 2$. Find the smallest positive integer $n$ such that $2^{2015}$ divides $m^n-1$.
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.
If $a+b=\sqrt{5}$, compute $\frac{a^2 -a^2b^2 + b^2 +2ab}{a+ab+b}$.
What is Heron's formula to calculate a triangle's area given the lengths of three sides?
(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$
What is the angle bisector's theorem?
Show that the next integer bigger than $(\sqrt{2}+1)^{2n}$ is divisible by $2^{n+1}$.
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.
As shown in the figure, in a regular triangular house, all the rooms are in the shape of regular triangles. There are doors between adjacent rooms. Starting from one of the rooms, go through the doors to visit other rooms, without repeating rooms or leaving the house. Including the starting room, how many rooms can be visited?
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 $$\left\{ \begin{array}{rcl} x &\equiv 2 &\pmod{3}\\ x &\equiv 2 &\pmod{5}\\ x &\equiv -3 &\pmod{7}\\x &\equiv -2 &\pmod{13} \end{array}\right.$$