Let $c$ be a nonzero real number. Suppose that $g(x)=c_0x^r+c_1x^{r-1}+\cdots+c_{r-1}x+c_r$ is a polynomial with integer coefficients. Suppose that the roots of $g(x)$ are $b_1,\cdots,b_r$. Let $k$ be a given positive integer. Show that there is a prime $p$ such that $p>\max(k,|c|,|c_r|)$, and moreover if $t$ is a real number between $0$ and $1$, and $j$ is one of $1,\cdots,r$, then \[|(\text{ }c^r\text{ }b_j\text{}g(tb_j)\text{ })^pe^{(1-t)b}|<\dfrac{(p-1)!}{2r}.\] Furthermore, if \[f(x)=\dfrac{e^{rp-1}x^{p-1}(g(x))^p}{(p-1)!}\] then \[\left|\sum_{j=1}^r\int_0^1 e^{(1-t)b_j}f(tb_j)dt\right|\leq \dfrac{1}{2}.\]

Let $x,y$ be the positive real numbers with $x+y=1$, and $n$ be the positive integer with $n\ge2$. Prove that \[\frac{x^n}{x+y^3}+\frac{y^n}{x^3+y}\ge\frac{2^{4-n}}{5}\]

There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.

For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.

Find all pairs of polynomials $p$, $q$ with complex coefficients so that \[p(x)\cdot q(x)=p(q(x)).\]

A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,$ \sqrt{\mbox{}}$) How can you find $ 3^{\sqrt{2}}$ with accuracy of 6 digits.

Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ , for all $x, y \in Z$, $x \ne 0$.

The product of the positive real numbers $x, y, z$ is 1. Show that if \[ \frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z \]then \[ \frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k} \] for all positive integers $k$.

Suppose $P(x) = x^{2016} + a_{2015}x^{2015} + ...+ a_1x + a_0$ satisfies $P(x)P(2x + 1) = P(-x)P(-2x - 1)$ for all $x \in R$. Find the sum of all possible values of $a_{2015}$.

Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that $$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$ for all nonnegative real numbers $x,y$ and $z$.

Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying: (i) there are only finitely many $s$ in ${\bf R}$ such that $f(s)=0$, and (ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.

Find all completely multipiclative functions $f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}$ such that for any $a,b\in \mathbb{Z}$ and $b\neq 0$, there exist integers $q,r$ such that $$a=bq+r$$ and $$f(r)<f(b)$$ Proposed by Navid Safaei

Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?

Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .

We say that a $2023$-tuple of nonnegative integers $(a_1,\hdots,a_{2023})$ is [i]sweet[/i] if the following conditions hold: [list] [*] $a_1+\hdots+a_{2023}=2023$ [*] $\frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1$ [/list] Determine the greatest positive integer $L$ so that \[a_1+2a_2+\hdots+2023a_{2023}\ge L\] holds for every sweet $2023$-tuple $(a_1,\hdots,a_{2023})$ [i]Ivan Novak[/i]

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously: $(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$; $(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?

Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\frac12$ , and otherwise, if he flips tails (T), he writes $-\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\frac14$ , and otherwise he writes $-\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \frac{1}{2n}$ , and otherwise he writes $- \frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\frac17$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.

Does there exist a triangle with length $a,b,c$ such that: [b]a)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$ [b]b)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$

Let $F(x_1,..., x_n)$ be a polynomial with real coefficients in $ n > 1$ “indeterminate” variables $x_1,..., x_n$. We say that $F$ is $n$-[i]alternating [/i]if for all integers $1 \le i < j \le n$, $$F(x_1,..., x_i,..., x_j,..., x_n) = - F(x_1,..., x_j,..., x_i,..., x_n),$$ i.e. swapping the order of indeterminates $x_i, x_j$ flips the sign of the polynomial. For example, $x^2_1x_2 - x^2_2x_1$ is $2$-alternating, whereas $x_1x_2x_3 +2x_2x_3$ is not $3$-alternating. [i]Note: two polynomials $P(x_1,..., x_n)$ and $Q(x_1,..., x_n)$ are considered equal if and only if each monomial constituent $ax^{k_1}_1... x^{k_n}_n$ of $P$ appears in $Q$ with the same coefficient $a$, and vice versa. This is equivalent to saying that $P(x_1,..., x_n) = 0$ if and only if every possible monomial constituent of $P$ has coefficient $0$. [/i] (1) Compute a $3$-alternating polynomial of degree $3$. (2) Prove that the degree of any nonzero $n$-alternating polynomial is at least ${n \choose 2}$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)^2 + |y|) = x^2 + f(y)$$

Let $n$ be a natural number. Solve the equation $$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying \[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\] [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]