Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

We call a polynomial $P$ [i]square-friendly[/i] if it is monic, has integer coefficients, and there is a polynomial $Q$ for which $P(n^2)=P(n)Q(n)$ for all integers $n$. We say $P$ is [i]minimally square-friendly[/i] if it is square-friendly and cannot be written as the product of nonconstant, square-friendly polynomials. Determine the number of nonconstant, minimally square-friendly polynomials of degree at most $12$.

Find all non-empty sets $S$ of nonzero real numbers such that a) $S$ has at most $5$ elements b) If $x$ is in $S$, then so are $1- x$ and $\frac{1}{x}$.

Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(f(x - y)) = f(x)f(y) + f(x) - f(y) - xy.\]

Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$

Maria was a brilliant mathematician who found the following property about her year of birth: if $f$ is a function defined in the set of natural numbers $N = \{0, 1, 2, 3, 4, 5,...\}$ such that $f(1) = 1335$ and $f(n+1) = f(n)-2n+43$ for all $n \in N$, then his year of birth is the maximum value that $f(n)$ can reach when $n$ takes values in $N$. Determine the year of birth of Mary.

For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows: $ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad \textbf{(B)}\ 2b^2 \equal{} 9ac\qquad \textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\ \textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad \textbf{(E)}\ 9b^2 \equal{} 2ac$

Find all reals $z$ such that $z^4 - z^3 - 2z^2 - 3z - 1= 0$.

Let $f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0 $ be a polynomial of degree $ n\ge 1 $ with $ n$ (not necessarily distinct) integer roots. Assume that there exist distinct primes $p_0,p_1,..,p_{n-1}$ such that $a_i > 1$ is a power of $p_i$, for all $ i=0,1,..,n-1$. Find all possible values of $ n$.

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$: (a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$ where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$

Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$. [I]Proposed by Italy[/i]

A monic polynomial is a polynomial whose highest degree coefficient is 1. Let $P(x)$ and $Q(x)$ be monic polynomial with real coefficients and $degP(x)=degQ(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions then $P(x+1)=Q(x-1)$ has a real solution

Determine the maximal value of $k$ such that the inequality $$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right) \le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$ holds for all positive reals $a, b, c$.

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables with distribution $\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12$. Let $Y_1$, $Y_2$, $Y_3$, and $Y_4$ be independent, identically distributed random variables, where $Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}$. Decide whether the random variables $Y_1+2Y_2+4Y_3+8Y_4$ and $Y_1+4Y_3$ are absolutely continuous.

Find all polynomials $P(x)$ with integral coefficients whose values at points $x = 1, 2, . . . , 2021$ are numbers $1, 2, . . . , 2021$ in some order.

Given positive real numbers $a,b,c$ with $ab+bc+ca\ge a+b+c$ , prove that $$(a + b + c)(ab + bc+ca) + 3abc \ge 4(ab + bc + ca).$$ (I. Gorodnin)

For $n = 1,2,3,...$. $a_n$ is defined by: $$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$ Prove that for every $n$ holds that $$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$

Decompose the polynomial $$x^8 + x^4 +1$$ to factors of at most second degree.

Find all pairs of real numbers $x$ and $y$ which satisfy the following equations: \begin{align*} x^2 + y^2 - 48x - 29y + 714 & = 0 \\ 2xy - 29x - 48y + 756 & = 0 \end{align*}

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, $\frac43$ seconds, $\frac53$ second and $2$ seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in $1$ minute? (Include first and last.)

Let $P(x)$ be a monic polynomial of degree $3$. Suppose that $P(x)$ has remainder $R(x)$ when it is divided by $(x - 1)(x - 4)$ and $2R(x)$ when it is divided by $(x - 2)(x - 3)$. Given that $P(0) = 5$, find $P(5)$.

Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.