Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

Determine the smallest real constant $C$ such that the inequality \[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\] holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$? [i](Walther Janous)[/i]

Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that: $(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$

A polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0$ with degree $d$ is [i]reflexive[/i] if there is an integer $n\ge d$ such that $c_i=c_{n-i}$ for every $0\le i\le n$, where $c_i=0$ for $i>d$. Let $\ell\ge 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x)$, $r(x)$ with integer coefficients such that \[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]

[u]Round 5[/u] [b]p13.[/b] Jason flips a coin repeatedly. The probability that he flips $15$ heads before flipping $4$ tails can be expressed as $\frac{a}{2^b}$ where $a$ and $b$ are positive integers and $a$ is odd. Find $a +b$. [b]p14.[/b] Triangle $ABC$ has side lengths $AB = 3$, $BC = 3$, and $AC = 4$. Let D be the intersection of the angle bisector of $\angle B AC$ and segment $BC$. Let the circumcircle of $\vartriangle B AD$ intersect segment $AC$ at a point $E$ distinct from $A$. The length of $AE$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p15.[/b] The sum of the squares of all values of $x$ such that $\{(x -2)(x -3)\} = \{(x -1)(x -6)\}$ and $\lfloor x^2 -6x +6 \rfloor= 9$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. Note: $\{a\}$ is the fractional part function, and returns $a -\lfloor a \rfloor$ . [u]Round 6[/u] [b]p16.[/b] Maisy the Polar Bear is at the origin of the Polar Plane ($r = 0, \theta = 0$). Maisy’s location can be expressed as $(r,\theta)$, meaning it is a distance of $r$ away from the origin and at a angle of $\theta$ degrees counterclockwise from the $x$-axis. When Maisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Maisy cannot jump to any point it has been to before. Let $L(r,\theta)$ be the number of paths Maisy can take to reach point $(r,\theta)$. The sum of $L(r,\theta)$ over all points where $r$ is an integer between $1$ and $2020$ and $\theta$ is an integer between $0$ and $359$ can be written as $\frac{n^k-1}{m}$ for some minimum value of $n$, such that $n$, $k$, and $m$ are all positive integers. Find $n +k +m$. [b]p17.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. Find $DF^2$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/0ee8716cebd6701fcae6544d9e39e68fff35f5.png[/img] [b]p18.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1} (2021i -1) = (2020)(4041)...(2021^2 -1).$$ [u]Round 7[/u] [b]p19.[/b] A function $f (n)$ is defined as follows: $$f (n) = \begin{cases} \frac{n}{3} \,\,\, if \,\,\, n \equiv 0 (mod \, 3) \\ n^2 +4n -5 \,\,\,if \,\,\,n \equiv 1 (mod \, 3) \\ n^2 +n -2 \,\,\, if \,\,\,n \equiv 2 (mod \, 3) \end{cases}$$ Find the number of integer values of $n$ between $2$ and $1000$ inclusive such that $f ( f (... f (n))) = 1$ for some number of applications of $f (n)$. [b]p20.[/b] In the diagram below, the larger circle with diameter $AW$ has radius $16$. $ABCD$ and $WXY Z$ are rhombi where $\angle B AD = \angle XWZ = 60^o$ and $AC = CY = YW$. $M$ is the midpoint of minor arc $AW$, as shown. Let $I$ be the center of the circle with diameter $OM$. Circles with center $P$ and $G$ are tangent to lines $AD$ and $WZ$, respectively, and also tangent to the circle with center $I$ . Given that $IP \perp AD$ and $IG \perp WZ$, the area of $\vartriangle PIG$ can be written as $a +b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of a prime. Find $a +b +c$. [b]p21.[/b] In a list of increasing consecutive positive integers, the first item is divisible by $1$, the second item is divisible by $4$, the third item is divisible by $7$, and this pattern increases up to the seventh item being divisible by $19$. Find the remainder when the least possible value of the first item in the list is divided by $100$. [u]Round 8[/u] [b]p22.[/b] Let the answer to Problem $24$ be $C$. Jacob never drinks more than $C$ cups of coffee in a day. He always drinks a positive integer number of cups. The probability that he drinks $C +1-X$ cups is $X$ times the probability he drinks $C$ cups of coffee for any positive number $X$ from $1$ to $C$ inclusive. Find the expected number of cups of coffee he drinks. [b]p23.[/b] Let the answer to Problem $22$ be $A$. Three lines are drawn intersecting the interior of a triangle with side lengths $26$, $28$, and $30$ such that each line is parallel and a distance A away from a respective side. The perimeter of the triangle formed by the three new lines can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p24.[/b] Let the answer to Problem $23$ be $B$. Given that $ab-c = bc-a = ca-b$ and $a^2+b^2+c^2 = B +2$, find the sum of all possible values of $|a +b +c|$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f(x)$ be a polynomial, such that $f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}$. Velly and Polly are taking turns, starting from Velly changing the coefficients $a_i$ with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?

Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$.

Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions: $f(x+y) < f(x) + f(y)$ $f(f(x)) = \lfloor {x} \rfloor + 2$

We define the ridiculous numbers recursively as follows: [list=a] [*]1 is a ridiculous number. [*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers. [/list] A closed interval $I$ is ``boring'' if [list] [*]$I$ contains no ridiculous numbers, and [*]There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers. [/list] The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations $$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$. (a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$; (b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

Let $a>0$ and $12a+5b+2c>0$. Prove that it is impossible for the equation $ax^2+bx+c=0$ to have two real roots in the interval $(2,3)$.

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

For real numbers $x \ge 0$ and $y \ge 0$, prove the inequality $$x^4+y^3+x^2+y+1 >\frac92 xy.$$

Let $P(x)$ a grade 3 polynomial such that: $$P(1)=1, P(2)=4, P(3)=9$$ Find the value of $P(10)+P(-6)$

Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\boxed{1} \ f(1) = 1$ $\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$

Consider the polynomials $P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x$ and $Q(x)=x^{4}-x^{3}-x^{2}-1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x)=0,$ find $P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).$

Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

Determine all functions $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for any positive reals $x,y$, $$f(xy+f(xy)) = xf(y) + yf(x)$$

Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that \[ (p(x)^2)'' \leq K(p'(x))^2. \]

Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$

Function $f: R\to R$ is said periodic , if $f$ is not a constant function and there is a number real positive $p$ with the property of $f (x) = f (x + p)$ for every $x \in R$. The smallest positive real number p which satisfies the condition $f (x) = f (x + p)$ for each $x \in R$ is named period of $f$. Given $a$ and $b$ real positive numbers, show that there are periodic functions $f_1$ and $f_2$, with periods $a$ and $b$ respectively, so that $f_1 (x)\cdot f_2 (x)$ is also a periodic function.

Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?