Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Let $n \geq 3$ be a positive integer. A [i]chipped $n$-board[/i] is a $2 \times n$ checkerboard with the bottom left square removed. Lino wants to tile a chipped $n$-board and is allowed to use the following types of tiles: [list] [*] Type 1: any $1 \times k$ board where $1 \leq k \leq n$ [*] Type 2: any chipped $k$-board where $1 \leq k \leq n$ that must cover the left-most tile of the $2 \times n$ checkerboard. [/list] Two tilings $T_1$ and $T_2$ are considered the same if there is a set of consecutive Type 1 tiles in both rows of $T_1$ that can be vertically swapped to obtain the tiling $T_2$. For example, the following three tilings of a chipped $7$-board are the same: [img]http://i.imgur.com/8QaSgc0.png[/img] For any positive integer $n$ and any positive integer $1 \leq m \leq 2n - 1$, let $c_{m,n}$ be the number of distinct tilings of a chipped $n$-board using exactly $m$ tiles (any combination of tile types may be used), and define the polynomial $$P_n(x) = \sum^{2n-1}_{m=1} c_{m,n}x^m.$$ Find, with justification, polynomials $f(x)$ and $g(x)$ such that $$P_n(x) = f(x)P_{n-1}(x) + g(x)P_{n-2}(x)$$ for all $n \geq 3$.

Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show that $x^3_1$ and $x^3_2$ are the roots of $$y^3-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3 =0.$$

Find all $f:R^+ \rightarrow R^+$ such that $f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$ @below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url] [quote]Feel free to start individual threads for the problems as usual[/quote]

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

$f(x)$ is square polynomial and $a \neq b$ such that $f(a)=b,f(b)=a$. Prove that there is not other pair $(c,d)$ that $f(c)=d,f(d)=c$

Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.

Let $ \{k_n\}_{n \equal{} 1}^{\infty}$ be a sequence of real numbers having the properties $ k_1 > 1$ and $ k_1 \plus{} k_2 \plus{} \cdots \plus{} k_n < 2k_n$ for $ n \equal{} 1,2,...$. Prove that there exists a number $ q > 1$ such that $ k_n > q^n$ for every positive integer $ n$.

How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$ $$ \mathrm a. ~ 180\qquad \mathrm b.~184\qquad \mathrm c. ~186 \qquad \mathrm d. ~189 \qquad \mathrm e. ~191 $$

Let f be a function defined in the set $\{0, 1, 2,...\}$ of non-negative integers, satisfying $f(2x) = 2f(x), f(4x+1) = 4f(x) + 3$, and $f(4x-1) = 2f(2x - 1) -1$. Show that $f $ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties: i) $P(0) = 1$ ii) for any real numbers $x$ and $y,$ \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\]

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$ When equality holds?

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.

If $ y\equal{}x\plus{}\frac{1}{x}$, then $ x^4\plus{}x^3\minus{}4x^2\plus{}x\plus{}1\equal{}0$ becomes: $ \textbf{(A)}\ x^2(y^2\plus{}y\minus{}2)\equal{}0 \qquad\textbf{(B)}\ x^2(y^2\plus{}y\minus{}3)\equal{}0\\ \textbf{(C)}\ x^2(y^2\plus{}y\minus{}4)\equal{}0 \qquad\textbf{(D)}\ x^2(y^2\plus{}y\minus{}6)\equal{}0\\ \textbf{(E)}\ \text{none of these}$

[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$. (a) Show that $\vartriangle DEF$ is also equilateral. (b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$. [b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color. [b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$ (a) Show that $ab$ cannot be the square of an integer. (b) Show that $ac$ cannot be the square of an integer. (c) Show that $abc$ cannot be the square of an integer. [b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$ $$ x^2+y^2=5$$ (a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection. [b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$. (a) Find $r(2, 3)$. (b) Find $r(n, 1)$. (c) Find, with justification, a formula for $r(n, m)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?

The real number $a>1$ is given. Suppose that $r$, $s$ and $t$ are different positive integer numbers such that $\{a^r\}=\{a^s\}=\{a^t\}$. Prove that $\{a^r\}=\{a^s\}=\{a^t\}=0$.

Real numbers $a,b,c,d$ satisfy the equality $$\frac{1-ab}{a+b}=\frac{bc-1}{b+c}=\frac{1-cd}{c+d}=\sqrt{3}$$ Find all possible values of $ad$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \] Holds for all $x \not= y \in \mathbb{R}$

Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .

Let $p, q,$ and $r$ be the roots of the polynomial $f(t) = t^3 - 2022t^2 + 2022t - 337.$ Given $$x = (q-1)\left ( \frac{2022 - q}{r-1} + \frac{2022 - r}{p-1} \right )$$ $$y = (r-1)\left ( \frac{2022 - r}{p-1} + \frac{2022 - p}{q-1} \right )$$ $$z = (p-1)\left ( \frac{2022 - p}{q-1} + \frac{2022 - q}{r-1} \right )$$ compute $xyz - qrx - rpy - pqz.$

Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find \[ \left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2. \]

Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.