[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The polynomial $P(x)=3x^3-2x^2+ax-b$ has roots $\sin^2\theta$, $\cos^2\theta$, and $\sin\theta\cos\theta$ for some angle $\theta$. Find $P(1)$.

The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.

In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $ 8$ and $ 2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $ \minus{}9$ and $ \minus{}1$ for the roots. The correct equation was: $ \textbf{(A)}\ x^2\minus{}10x\plus{}9\equal{}0 \qquad\textbf{(B)}\ x^2\plus{}10x\plus{}9\equal{}0 \qquad\textbf{(C)}\ x^2\minus{}10x\plus{}16\equal{}0\\ \textbf{(D)}\ x^2\minus{}8x\minus{}9\equal{}0 \qquad\textbf{(E)}\ \text{none of these}$

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 2 Let $A$, $B$ and $C$ be real numbers such that (i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$, (ii) $\tan C$ and $\cot C$ are defined. Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.

Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

For which natural numbers $n$ can the polynomial $f (x) = x^n + x^{n-1} +...+ x + 1$ as write $f (x) = g (h (x))$, where $g$ and $h$ should be real polynomials of degrees greater than $1$?

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ . .

How many solutions does \[ x^2 - [x^2] = \left(x - [x]\right)^2 \] have satisfying $1 \leq x \leq n$?

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positives reals $a_1,a_2,\cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0=x$, and $x_k=\sqrt{x_{k-1}a_k}$ for all $1\leq k\leq n$. Finally, let $y=x_n$. We said $M>0$ is "tremendous" if for any $x\in \mathbb{R}^+$, we can always choose $n,a_1,a_2,\cdots, a_n$ to make the resulting $y$ smaller than $M$. Find all tremendous numbers. [i]Proposed by ckliao914.[/i]

a,b,c>0 and $abc\ge 1$.Prove that: $\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$

Let $\{x_n\}$ be a Van Der Corput series,that is,if the binary representation of $n$ is $\sum a_{i}2^{i}$ then $x_n=\sum a_i2^{-i-1}$.Let $V$ be the set of points on the plane that have the form $(n,x_n)$.Let $G$ be the graph with vertex set $V$ that is connecting any two points $(p,q)$ if there is a rectangle $R$ which lies in parallel position with the axes and $R\cap V= \{p,q\}$.Prove that the chromatic number of $G$ is finite.

Three non-zero real numbers $a, b, c$ satisfy $a + b + c = 0$ and $a^4 + b^4 + c^4 = 128$. Determine all possible values of $ab + bc + ca$.

Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_0<s_1<s_2<\cdots$ satisfying the following properties: [list=disc] [*]the sequence $s_1-s_0$, $s_2-s_1$, $s_3-s_2$, $\ldots$ is periodic; and [*]$s_{s_n}-s_{s_{n-1}}\leq N<s_{1+s_n}-s_{s_{n-1}}$ for all positive integers $n$ [/list]

Find all ordered 4-tuples of integers $(a,b,c,d)$ (not necessarily distinct) satisfying the following system of equations: \begin{align*}a^2-b^2-c^2-d^2&=c-b-2\\2ab&=a-d-32\\2ac&=28-a-d\\2ad&=b+c+31.\end{align*}

For positive integers $j\le n$, prove that $$\sum_{k=j}^n\binom{2n}{2k}\binom kj=\frac{n\cdot4^{n-j}}j\binom{2n-j-1}{j-1}.$$ [i]Proposed by Ángel Plaza[/i]

Suppose that \[S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}\] Find $\lim_{n \to \infty} S_n.$

Find all real solutions of the system of equations $$\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 \\ xu + yv + xv + yu = 0 \\ xyu + yuv + uvx + vxy = - 2 \\ xyuv = -1 \end{cases}$$

Let $\lfloor x\rfloor$ denote the greatest integer function and $\{x\}=x-\lfloor x\rfloor$ denote the fractional part of $x$. Let $1\leq x_1<\ldots<x_{100}$ be the $100$ smallest values of $x\geq 1$ such that $\sqrt{\lfloor x\rfloor\lfloor x^3\rfloor}+\sqrt{\{x\}\{x^3\}}=x^2.$ Compute \[\sum_{k=1}^{50}\dfrac{1}{x_{2k}^2-x_{2k-1}^2}.\]

Let $a,b,c,\alpha,\beta,\gamma \in\mathbb{R}$ such as $a^2+b^2+c^2 \neq 0 \neq \alpha\beta\gamma$ and $24^{\alpha}\neq 3^{\beta} \neq 2012^{\gamma} \neq 24^{\alpha}$. Prove that the equation \[ a \cdot 24^{\alpha x}+b \cdot 3^{\beta x} + c \cdot 2012^{\gamma x}=0 \] has at most two real solutions.

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$? $ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $

Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$: (a) $f (n)$ is a positive integer, (b) $f (n + 1) > f (n)$, (c) $f ( f (n)) = 3n$. Find $f (2001)$.