Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. (a) Let $p$ be a prime. Consider the graph whose vertices are the ordered pairs $(x, y)$ with $x, y \in\{0, 1, . . . , p-1\}$ and whose edges join vertices $(x, y)$ and $(x' , y')$ if and only if $xx'+yy'\equiv 1 \pmod{p}$ . Prove that this graph does not contain $C_{4}$ . (b) Prove that for infinitely many values $n$ there is a graph $G_{n}$ with $e(G_{n}) \geq \frac{n\sqrt{n}}{2}-n$ that does not contain $C_{4}$.

Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that $$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$ Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties: (a) $f(1) = 0$, (b) $f(x) > 0$ for all $x > 1$, (c) For all $x, y\ge 0$ with $x + y > 0$ holds $$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$

For a given positive integer $n,$ find $$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$

Solve in the reals the equation $ 2^{\lfloor\sqrt[3]{x}\rfloor } =x. $ [i]Nedelcu Ion[/i]

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

Let $n \ge 2$ be a positive integer. Suppose that $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ are 2n numbers such that $\sum_{i=1}^n a_i =\sum_{i=1}^n n_i= 1$ and $a_i\ge 0, 0 \le b_i\le \frac{n-1}{n}, i = 1, 2,..., n$. Show that $$b_1a_2a_3...a_n+a_1b_2a_3...a_n+...+a_1a_2...a_{k-1}b_ka_{k+1}...a_n+ ...+ a_1a_2...a_{n-1}b_n \le \frac{1}{n(n-1)^{n-2}}$$

Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]

Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$. We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?

Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$

[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$? [b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails? [b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$. [b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$? [b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average. [b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number. [b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle. [b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping. [b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$. [b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ : $P(x_{i},y_{i})=b_{i}$

Jack writes whole numbers starting from $ 1$ and skips all numbers that contain either a $2$ or $9$. What is the $100$th number that Jack writes down?

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]

Let $n$ and $k$ be positive integers. Find all monic polynomials $f\in \mathbb{Z}[X]$, of degree $n$, such that $f(a)$ divides $f(2a^k)$ for $a\in \mathbb{Z}$ with $f(a)\neq 0$.

Dene the function $f : R \to R$ by $$f(x) =\begin{cases} \dfrac{1}{x^2+\sqrt{x^4+2x}}\,\,\, \text{if} \,\,\,x \notin (- \sqrt[3]{2}, 0] \\ \,\,\, 0 \,\,\,, \,\,\, \text{otherwise} \end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x) = 1$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a, b,c, d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d.$ (Here, $f^n(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^3(x) = f(f(f(x)))$.)

Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point ${z}$, satisfy $$\frac {1+z^{23}}{z^{64}}\in\mathbb R.$$ (2) For any non-zero complex number ${a}$ and any real number $\theta$, the equation $1+z^{23}+az^{64}=0$ has roots in $$S_{\theta}=\left\{ z\in\mathbb C\mid\operatorname{Re}(ze^{-i\theta })\geqslant |z|\cos\frac{\pi}{20}\right\}.$$ [i]Proposed by Yijun Yao[/i]

Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$ a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients. b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.

Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds: \[ (x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x). \]

Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.

For natural numbers $x_1,\ldots,x_k$, let $[x_k,\ldots,x_1]$ be defined recurrently as follows: $[x_2,x_1]=x_2^{x_1}$ and, for $k\ge3$, $[x_k,x_{k-1},\ldots,x_1]=x_k^{[x_{k-1},\ldots,x_1]}$. (a) Let $3\le a_1\le a_2\le\ldots\le a_n$be integers. For a permutation $\sigma$ of the set $\{1,2,\ldots,n\}$, we set $P(\sigma)=[a_{\sigma(n)},a_{\sigma(n-1)},\ldots,a_{\sigma(1)}]$. Find the permutations $\sigma$ for which $P(\sigma)$ is minimal or maximal. (b) Find all integers $a,b,c,d$, greater than or equal to $2$, for which $[178,9]\le[a,b,c,d]\le[198,9]$.

a) Prove that for any nonnegative real $x$, holds $$x^{\frac32} + 6x^{\frac54} + 8x^{\frac34}\ge 15x.$$ b) Determine all x for which the equality holds