We are given a natural number $d$. Find all open intervals of maximum length $I \subseteq R$ such that for all real numbers $a_0,a_1,...,a_{2d-1}$ inside interval $I$, we have that the polynomial $P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0$ has no real roots.

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

Let $n \in \mathbb{N}_{\geq 2}.$ For any real numbers $a_1,a_2,...,a_n$ denote $S_0=1$ and for $1 \leq k \leq n$ denote $$S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k}$$ Find the number of $n-$tuples $(a_1,a_2,...a_n)$ such that $$(S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n.$$

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that \[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\] holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.

Let $x_1,x_2,... ,x_{2014}$ be real numbers different from $1$, such that $x_1 + x_2 +...+x_{2014} = 1$ and $\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1$ What is the value of $\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}$ ?

Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$

Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$. Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that $|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$, with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

The number of triples $(a,b,c)$ of positive integers which satisfy the simultaneous equations \begin{align*} ab+bc &= 44,\\ ac+bc &= 23, \end{align*} is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals: $\text{(A)}\ \dfrac{5}{2}\qquad \text{(B)}\ \frac{3}{2}\qquad \text{(C)}\ \dfrac{4}{3}\qquad \text{(D)}\ \dfrac{5}{4}\qquad \text{(E)}\ \dfrac{3}{4}$

Positive integers $x, y$ satisfy the following conditions: $$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$ Show that $x = y$. Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$. [i]Proposed by Anton Trygub[/i]

Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point. EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet. Note: fresh translation

Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$

A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two. How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic? (Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as \[A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}\] respectively, where $H(a,b)$ is not defined for some $a$, $b$.)

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true: $$f(x+y+f(y))=f(x)+2y$$ [I]proposed by F. Petrov[/i]

Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality : $${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$

Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.

[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$? [b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$. [b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit? [b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$. [b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property. [b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$? [b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$. [b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

Find out which of the two numbers is greater: $$\dfrac{2}{2 +\dfrac{2}{2 +\dfrac{2}{... +\dfrac{2}{2+\frac22}}}} \,\,\, \text{or} \,\,\, \dfrac{3}{3 +\dfrac{3}{3 +\dfrac{3}{... +\dfrac{3}{3+\frac33}}}}$$ (Each expression has $2022$ fraction signs.)