Three strictly increasing sequences \[a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots\] of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold: (a) $c_{a_n}=b_n+1$; (b) $a_{n+1}>b_n$; (c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even. Find $a_{2010}$, $b_{2010}$ and $c_{2010}$. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 1)[/i]

A polynomial $f(x)$ with real coefficients is called [i]generating[/i], if for each polynomial $\varphi(x)$ with real coefficients there exists a positive integer $k$ and polynomials $g_1(x),\dotsc,g_k(x)$ with real coefficients such that \[\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).\] Find all generating polynomials.

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded. [i]Proposed by Palmer Mebane, United States[/i]

Determine all triples of positive real numbers $(x, y, z)$ such that \[x+y+z=6,\]\[\frac 1x + \frac 1y + \frac 1z = 2 - \frac{4}{xyz}.\]

Prove that $x^{12} - x^9 + x^4 - x + 1 > 0$ for all $x$.

[b]p1.[/b] Given that $7 \times 22 \times 13 = 2002$, compute $14 \times 11 \times 39$. [b]p2.[/b] Ariel the frog is on the top left square of a $8 \times 10$ grid of squares. Ariel can jump from any square on the grid to any adjacent square, including diagonally adjacent squares. What is the minimum number of jumps required so that Ariel reaches the bottom right corner? [b]p3.[/b] The distance between two floors in a building is the vertical distance from the bottom of one floor to the bottom of the other. In Evans hall, the distance from floor $7$ to floor $5$ is $30$ meters. There are $12$ floors on Evans hall and the distance between any two consecutive floors is the same. What is the distance, in meters, from the first floor of Evans hall to the $12$th floor of Evans hall? [b]p4.[/b] A circle of nonzero radius $ r$ has a circumference numerically equal to $\frac13$ of its area. What is its area? [b]p5.[/b] As an afternoon activity, Emilia will either play exactly two of four games (TwoWeeks, DigBuild, BelowSaga, and FlameSymbol) or work on homework for exactly one of three classes (CS61A, Math 1B, Anthro 3AC). How many choices of afternoon activities does Emilia have? [b]p6.[/b] Matthew wants to buy merchandise of his favorite show, Fortune Concave Decagon. He wants to buy figurines of the characters in the show, but he only has $30$ dollars to spend. If he can buy $2$ figurines for $4$ dollars and $5$ figurines for $8$ dollars, what is the maximum number of figurines that Matthew can buy? [b]p7.[/b] When Dylan is one mile from his house, a robber steals his wallet and starts to ride his motorcycle in the direction opposite from Dylan’s house at $40$ miles per hour. Dylan dashes home at $10$ miles per hour and, upon reaching his house, begins driving his car at $60$ miles per hour in the direction of the robber’s motorcycle. How long, starting from when the robber steals the wallet, does it take for Dylan to catch the robber? Express your answer in minutes. [b]p8.[/b] Deepak the Dog is tied with a leash of $7$ meters to a corner of his $4$ meter by $6$ meter rectangular shed such that Deepak is outside the shed. Deepak cannot go inside the shed, and the leash cannot go through the shed. Compute the area of the region that Deepak can travel to. [img]https://cdn.artofproblemsolving.com/attachments/f/8/1b9563776325e4e200c3a6d31886f4020b63fa.png[/img] [b]p9.[/b] The quadratic equation $a^2x^2 + 2ax -3 = 0$ has two solutions for x that differ by $a$, where $a > 0$. What is the value of $a$? [b]p10.[/b] Find the number of ways to color a $2 \times 2$ grid of squares with $4$ colors such that no two (nondiagonally) adjacent squares have the same color. Each square should be colored entirely with one color. Colorings that are rotations or reflections of each other should be considered different. [b]p11[/b]. Given that $\frac{1}{y^2+5} - \frac{3}{y^4-39} = 0$, and $y \ge 0$, compute $y$. [b]p12.[/b] Right triangle $ABC$ has $AB = 5$, $BC = 12$, and $CA = 13$. Point $D$ lies on the angle bisector of $\angle BAC$ such that $CD$ is parallel to $AB$. Compute the length of $BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d5cddb0e8ac43c35ddfc94b2a74b8d022292f2.png[/img] [b]p13.[/b] Let $x$ and $y$ be real numbers such that $xy = 4$ and $x^2y + xy^2 = 25$. Find the value of $x^3y +x^2y^2 + xy^3$. [b]p14.[/b] Shivani is planning a road trip in a car with special new tires made of solid rubber. Her tires are cylinders that are $6$ inches in width and have diameter $26$ inches, but need to be replaced when the diameter is less than $22$ inches. The tire manufacturer says that $0.12\pi$ cubic inches will wear away with every single rotation. Assuming that the tire manufacturer is correct about the wear rate of their tires, and that the tire maintains its cylindrical shape and width (losing volume by reducing radius), how many revolutions can each tire make before she needs to replace it? [b]p15.[/b] What’s the maximum number of circles of radius $4$ that fit into a $24 \times 15$ rectangle without overlap? [b]p16.[/b] Let $a_i$ for $1 \le i \le 10$ be a finite sequence of $10$ integers such that for all odd $i$, $a_i = 1$ or $-1$, and for all even $i$, $a_i = 1$, $-1$, or $0$. How many sequences a_i exist such that $a_1+a_2+a_3+...+a_{10} = 0$? [b]p17.[/b] Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ such that $AB$ and $BC$ have integer side lengths. Squares $ABDE$ and $BCFG$ lie outside $\vartriangle ABC$. If the area of $\vartriangle ABC$ is $12$, and the area of quadrilateral $DEFG$ is $38$, compute the perimeter of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/6/980d3ba7d0b43507856e581476e8ad91886656.png[/img] [b]p18.[/b] What is the smallest positive integer $x$ such that there exists an integer $y$ with $\sqrt{x} +\sqrt{y} = \sqrt{1025}$ ? [b]p19. [/b]Let $a =\underbrace{19191919...1919}_{19\,\, is\,\,repeated\,\, 3838\,\, times}$. What is the remainder when $a$ is divided by $13$? [b]p20.[/b] James is watching a movie at the cinema. The screen is on a wall and is $5$ meters tall with the bottom edge of the screen $1.5$ meters above the floor. The floor is sloped downwards at $15$ degrees towards the screen. James wants to find a seat which maximizes his vertical viewing angle (depicted below as $\theta$ in a two dimensional cross section), which is the angle subtended by the top and bottom edges of the screen. How far back from the screen in meters (measured along the floor) should he sit in order to maximize his vertical viewing angle? [img]https://cdn.artofproblemsolving.com/attachments/1/5/1555fb2432ee4fe4903accc3b74ea7215bc007.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy: $\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.

The plane is divided into domains by $ n$ straight lines in general position, where $ n \geq 3$. Determine the maximum and minimum possible number of angular domains among them. (We say that $ n$ lines are in general position if no two are parallel and no three are concurrent.)

The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\] Show that $ a_{2008} < 5$.

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

The sequence $\{x_n\}_{n=0}^\infty$ is defined by the following equations: $x_n=\sqrt{x_{n-1} x_{n-2}+\frac{n}{2}}$ ,$\forall$ $n\geq 2$, $x_0=x_1=1$. Prove that there exist a real number $a$, such that $an<x_n<an+1$ for each natural number $n$.

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

Show that for all integers $ n \ge 2$, $ \sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2$

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that \[|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.\] Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$

Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.

Let $Z\subset \mathbb{C}$ be a set of $n$ complex numbers, $n\geqslant 2.$ Prove that for any positive integer $m$ satisfying $m\leqslant n/2$ there exists a subset $U$ of $Z$ with $m$ elements such that\[\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.\][i]Vasile Pop[/i]

For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

Let $(a_n)$ and $(b_n)$ be sequences of real numbers, such that $a_1 = b_1 = 1$, $a_{n+1} = a_n + \sqrt{a_n}$, $b_{n+1} = b_n + \sqrt[3]{b_n}$ for all positive integers $n$. Prove that there is a positive integer $n$ for which the inequality $a_n \leq b_k < a_{n+1}$ holds for exactly 2021 values of $k$.

For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$

Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$

Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.

Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$ satisfy $|z_3|=|z_4|=1$

Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le $1 and $x^5 + y^5 \le 1$. Prove that $a^2x^3 + b^2y^3 \le 1$.

Let n be an integer which is greater than 1, not divisible by 1997. Let $ a_m\equal{}m\plus{}\frac{mn}{1997}$ for all m=1,2,..,1996 $ b_m\equal{}m\plus{}\frac{1997m}{n}$ for all m=1,2,..,n-1 We arrange the terms of two sequence $ (a_i), (b_j)$ in the ascending order to form a new sequence $ c_1\le c_2\le ...\le c_{1995\plus{}n}$ Prove that $ c_{k\plus{}1}\minus{}c_k<2$ for all k=1,2,...,1994+n