Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).

A polynomial $P$ with real coefficients is called [i]great,[/i] if for some integer $a>1$ and for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all [i]great[/i] polynomials. [i]Proposed by A. Golovanov[/i]

A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane. [i]P.A.Kozhevnikov[/i]

Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that \[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]

Let $n$ be the $200$th smallest positive real solution to the equation $x- \frac{\pi}{2} =\ tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.

Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$.

For each positive integer $n$ let the set $A_n$ consist of all numbers $\pm 1 \pm 2 \pm ...\pm n$. For example, $$A_1 = \{-1,1\}, A_2 = \{ -3 ,-1 ,1 ,3 \} , A_3 = \{ -6 ,-4 ,-2 ,0 ,2 ,4 ,6 \}.$$ Find the number of elements in $A_n$ .

If $a, b, c$ and $d$ are complex non-zero numbers such that $$2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|.$$ Prove that $$\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .$$

Let $p$ and $q$ be two distinct integers. The square trinomial $x^2 + px + q$ is written on the board. At each step, a number is deleted: or the coefficient next to $x$, or the free term, and instead of the deleted number, a number is written, which is obtained from the deleted number by adding or subtracting the number $1$. After several such steps on the board, the square trinomial $x^2 + qx + p$ appeared. Show that at one stage a square trinomial was written on the board, both roots of which are integers.

In how many different ways can one select two distinct subsets of the set $\{1,2,3,4,5,6,7\}$, so that one includes the other? $ \textbf{(A)}\ 2059 \qquad \textbf{(B)}\ 2124 \qquad \textbf{(C)}\ 2187 \qquad \textbf{(D)}\ 2315 \qquad \textbf{(E)}\ 2316$

Given a polynomial $P\in \mathbb{Z}[X]$ of degree $k$, show that there always exist $2d$ distinct integers $x_1, x_2, \cdots x_{2d}$ such that $$P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d})$$ for some $d\le k+1$. [Extra: Is this still true if $d\le k$? (Of course false for linear polynomials, but what about higher degree?)]

A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .

Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$. [i]Marius Cavachi[/i]

Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

Let $a_0=0$, $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers. (i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which $$p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k$$ and $$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)$$ where the constant $c$ depends only on the numbers $a_i, b_i$. (ii) Prove that, in general, (*) cannot be replaced by the relation $$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$ [J. Szabados]

Show that for any real numbers $x, y > 1$, we have $$\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$$

Find the equation of cylinder that passes three straight lines $L_1= \begin{cases} x=0\\ y-z=2 \end{cases}, L_2= \begin{cases} x=0\\ x+y-z+2=0 \end{cases}, L_3= \begin{cases} x=\sqrt{2}\\ y-z=0 \end{cases}$.

[b]p1.[/b] For positive real numbers $x, y$, the operation $\otimes$ is given by $x \otimes y =\sqrt{x^2 - y}$ and the operation $\oplus$ is given by $x \oplus y =\sqrt{x^2 + y}$. Compute $(((5\otimes 4)\oplus 3)\otimes2)\oplus 1$. [b]p2.[/b] Janabel is cutting up a pizza for a party. She knows there will either be $4$, $5$, or $6$ people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices? [b]p3.[/b] If the numerator of a certain fraction is added to the numerator and the denominator, the result is $\frac{20}{19}$ . What is the fraction? [b]p4.[/b] Let trapezoid $ABCD$ be such that $AB \parallel CD$. Additionally, $AC = AD = 5$, $CD = 6$, and $AB = 3$. Find $BC$. [b]p5.[/b] AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream? [b]p6.[/b] Find the minimum possible value of the expression $|x+1|+|x-4|+|x-6|$. [b]p7.[/b] How many $3$ digit numbers have an even number of even digits? [b]p8.[/b] Given that the number $1a99b67$ is divisible by $7$, $9$, and $11$, what are $a$ and $b$? Express your answer as an ordered pair. [b]p9.[/b] Let $O$ be the center of a quarter circle with radius $1$ and arc $AB$ be the quarter of the circle’s circumference. Let $M$,$N$ be the midpoints of $AO$ and $BO$, respectively. Let $X$ be the intersection of $AN$ and $BM$. Find the area of the region enclosed by arc $AB$, $AX$,$BX$. [b]p10.[/b] Each square of a $5$-by-$1$ grid of squares is labeled with a digit between $0$ and $9$, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by $3$. How many such labelings are possible if each digit can be used more than once? [b]p11.[/b] A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is $5$, how many different possible values of the units digit are there? [b]p12.[/b] There are $2019$ red balls and $2019$ white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red? [b]p13.[/b] Let $ABCD$ be a square with side length $2$. Let $\ell$ denote the line perpendicular to diagonal $AC$ through point $C$, and let $E$ and $F$ be themidpoints of segments $BC$ and $CD$, respectively. Let lines $AE$ and $AF$ meet $\ell$ at points $X$ and $Y$ , respectively. Compute the area of $\vartriangle AXY$ . [b]p14.[/b] Express $\sqrt{21-6\sqrt6}+\sqrt{21+6\sqrt6}$ in simplest radical form. [b]p15.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length two. Let $D$ and $E$ be on $AB$ and $AC$ respectively such that $\angle ABE =\angle ACD = 15^o$. Find the length of $DE$. [b]p16.[/b] $2018$ ants walk on a line that is $1$ inch long. At integer time $t$ seconds, the ant with label $1 \le t \le 2018$ enters on the left side of the line and walks among the line at a speed of $\frac{1}{t}$ inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when $t = 2019$ seconds. [b]p17.[/b] Determine the number of ordered tuples $(a_1,a_2,... ,a_5)$ of positive integers that satisfy $a_1 \le a_2 \le ... \le a_5 \le 5$. [b]p18.[/b] Find the sum of all positive integer values of $k$ for which the equation $$\gcd (n^2 -n -2019,n +1) = k$$ has a positive integer solution for $n$. [b]p19.[/b] Let $a_0 = 2$, $b_0 = 1$, and for $n \ge 0$, let $$a_{n+1} = 2a_n +b_n +1,$$ $$b_{n+1} = a_n +2b_n +1.$$ Find the remainder when $a_{2019}$ is divided by $100$. [b]p20.[/b] In $\vartriangle ABC$, let $AD$ be the angle bisector of $\angle BAC$ such that $D$ is on segment $BC$. Let $T$ be the intersection of ray $\overrightarrow{CB}$ and the line tangent to the circumcircle of $\vartriangle ABC$ at $A$. Given that $BD = 2$ and $TC = 10$, find the length of $AT$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.

Suppose $a_3x^3 - x^2 + a_1x - 7 = 0$ is a cubic polynomial in x whose roots $\alpha,\beta, \gamma$ are positive real numbers satisfying $$\frac{225\alpha^2}{\alpha^2 +7}=\frac{144\beta^2}{\beta^2 +7}=\frac{100\gamma^2}{\gamma^2 +7}.$$ Find $a_1$.

Find all real solutions of the equation $(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3$.

A sequence $(a_n)_{n\ge0}$ satisfies $a_{m+n}+a_{m-n}=\frac12\left(a_{2m}+a_{2n}\right)$ for all integers $m,n$ with $m\ge n\ge0$. Given that $a_1=1$, find $a_{2003}$.