A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC \cdot DA.$ Point $X$ lies inside $ABCD$ so that $\angle XAB = \angle XCD$ and $\angle XBC = \angle XDA.$ Prove that $\angle BXA + \angle DXC = 180^{\circ}$ .

Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. Find the probability that she will roll the die an odd number of times.

Solve $x^{22} + x^{11}\equiv 2\pmod{11}$.

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.

Show that a prime $p > 2$ is a sum of two squares if and only if $p\equiv 1\pmod{4}$.

(Two Squares Theorem) Show that a positive integer $n$ is a sum of two squares if and only if each prime factor $p$ of $n$ such that $p\equiv 3\pmod{4}$ occurs to an even power in the prime factorization of $n$.

Let $n$ be an odd integer greater than $3$, and $\mathbb{S}=\{0, 1, \cdots, n-1\}$. Show that after removing any element from $\mathbb{S}$, it is always possible to equally divide the remaining elements in $\mathbb{S}$ into two groups such that their sum are congruent modulo $n$.

Let $n$ be a positive integer not less than $4$. Show that there exists a polynomial with integral coefficients $$f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2}+\cdots + a_1 x + a_0$$

such that for any positive integer $m$ and any $k \ge 2$ distinct integers $r_1$, $r_2$, $\cdots$, $r_k$, it always hold that $f(m)\ne f(r_1)f(r_2)\cdots f(r_k)$.

Let $p$ be a prime and $$\frac{a}{b}=\frac{1}{1^2}+\frac{1}{2^2}+\cdots + \frac{1}{(p-1)^2}$$

where $a$ and $b$ are two co-prime positive integers. Show that $p\mid a$.

(Fermat's little theorem) Show that $a^p\equiv a\pmod{p}$ holds if $p$ is a prime.

Find all powers of $2$, such that after deleting its first digit, the new number is also a power of 2. For example, $32$ is such a number because $32=2^5$ and $2=2^1$.

An integer in the form of $F_n=2^{2^n}+1$ where integer $n\ge 1$ is called a Fermat's number. Let $d_n$ be any divisor of $F_n$. Show that $d_n\equiv 1\pmod{2^{n+1}}$.

Assume positive integer $n > 1$ satisfies $n\mid (2^n+1)$, prove $n$ is a multiple of $3$.

Let $p$ be an odd prime, and $n=\frac{2^{2p}-1}{3}$ in an integer. Prove $2^{n-1}\equiv 1\pmod{n}$.

Show that there exists an infinite number of squares in the form of $(n\cdot 2^k - 7)$ where $n$ and $k$ are both positive integers.

Show that the number $(2n^{3k}+4n^{k}+10)$ cannot be a product of consecutive integers for any positive integers $n$ and $k$.

Let integers $l > m > n$ be the side lengths of a triangle satisfying $\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ where function $\{x\}$ returns the decimal part of real number $x$. Find the least possible value of this triangle's perimeter.

Let $p$, $q$, and $n$ be three positive integers, show that $$\sum_{k=0}^n\binom{p+k}{p}\binom{q+n-k}{q} = \binom{p+q+n+1}{p+q+1}$$

Let $\lfloor{x}\rfloor$ be the largest integer not exceeding real number $x$. Show that $$\sum_{k=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\left(\left(1-\frac{2k}{n}\right)\binom{n}{k}\right)^2=\frac{1}{n}\binom{2n-2}{n-1}$$

Let $n$ be a positive integer and function $\lfloor{x}\rfloor$ return the largest integer not exceeding $x$. Compute the value of $$\sum_{k=0}^{\lfloor{\frac{n}{2}}\rfloor}\binom{n-k}{k}$$

Show that $$\sum_{k=0}^{n}(-1)^k2^{2n-2k}\binom{2n-k+1}{k}=n+1$$

Show that $$\sqrt{1+x}=1+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n\cdot 2^{2n-1}}\binom{2n-2}{n-1}x^n$$

Show that $$\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty}\binom{2n}{n}x^n$$

(Generalized binomial expansion) If $a$, $b$, and $r$ are some real or complex numbers, then $$(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$$

Here, the following definition still holds when $r$ is a real or complex number: $$\binom{r}{k}=\frac{r(r-1)\cdots(r-k+1)}{1\cdot 2\cdots k}$$