The microcalculator ''MK-97'' can work out the numbers entered in memory, perform only three operations: a) check whether the selected two numbers are equal; b) add the selected numbers; c) using the selected numbers $a$ and $b$, find the equation $x^2 +ax+b = 0$, and if there are no roots, display a message about this. The results of all actions are stored in memory. Initially, one number $x$ is stored in memory. How to use ''MK-97'' to find out whether is this number one?

Solve the system of equations: $ (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1}$ $ y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0$

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of $1000$.

Find all real triples $(a,b,c)$ satisfying \[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]

Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

$A=\{1,2,3,4,5,6,7,8\} .$ How many functions $f: A \rightarrow A$ are there such that $f(1)<f(2)<f(3)$ [list=1] [*] ${{8}\choose{3}}$ [*] ${{8}\choose{3}}5^{8}$ [*] ${{8}\choose{3}} 8^{5}$ [*] $\frac{8 !}{3 !} $ [/list]

Given two rational numbers $ a,b, $ find the functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify $$ f(x+a+f(y))=f(x+b)+y, $$ for any rational $ x,y. $ [i]Vasile Pop[/i]

[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. [b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively [b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b] [b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img] [b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

The number 104,060,465 is divisible by a five-digit prime number. What is that prime number?

Let $a, b, c$ be positive numbers with $\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}$. Prove that the system of equations \[\sqrt{y-a}+\sqrt{z-a}=1,\] \[\sqrt{z-b}+\sqrt{x-b}=1,\] \[\sqrt{x-c}+\sqrt{y-c}=1\] has exactly one solution $(x, y, z)$ in real numbers.

Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$. Find the maximum possible value of $n$. Further characterise all such maximal configurations.

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

Solve the system $\begin {cases} 16x^3 + 4x = 16y + 5 \\ 16y^3 + 4y = 16x + 5 \end{cases}$

Let $n \in \mathbb{N}^+,$ $x_1,x_2,...,x_{n+1},p,q\in \mathbb{R}^+ $ , $p<q$ and $x^p_{n+1}>\sum_{i=1}^{n}x^p_{i}.$ Prove that $(1)x^q_{n+1}>\sum_{i=1}^{n}x^q_{i};$ $(2)\left(x^p_{n+1}-\sum_{i=1}^{n}x^p_{i}\right)^{\frac{1}{p}}<\left(x^q_{n+1}-\sum_{i=1}^{n}x^q_{i}\right)^{\frac{1}{q}}.$

Let be three finite and nonempty sets $ A,B,C $ such that $ |A|=|C|\le |B| , $ and a bijection $ A\stackrel{\beta }{\longrightarrow } C. $ How many pairs of functions $ A\stackrel{f_2 }{\longrightarrow } B\stackrel{f_1 }{\longrightarrow } C $ that satisfy $ f_1\circ f_2=\beta $ are there?

[u]Round 5[/u] [b]p13.[/b] Two concentric circles have radii $1$ and $3$. Compute the length of the longest straight line segment that can be drawn froma point on the circle of radius $1$ to a point on the circle of radius $3$ if the segment cannot intersect the circle of radius $1$. [b]p14.[/b] Find the value of $\frac{1}{3} + \frac29+\frac{3}{27}+\frac{4}{81}+\frac{5}{243}+...$ [b]p15.[/b] Bob is trying to type the word "welp". However, he has a $18$ chance ofmistyping each letter and instead typing one of four adjacent keys, each with equal probability. Find the probability that he types "qelp" instead of "welp". [u]Round 6[/u] [b]p16.[/b] How many ways are there to tile a $2\times 12$ board using an unlimited supply of $1\times 1$ and $1\times 3$ pieces? [b]p17.[/b] Jeffrey and Yiming independently choose a number between $0$ and $1$ uniformly at random. What is the probability that their two numbers can formthe sidelengths of a triangle with longest side of length $1$? [b]p18.[/b] On $\vartriangle ABC$ with $AB = 12$ and $AC = 16$, let $M$ be the midpoint of $BC$ and $E$,$F$ be the points such that $E$ is on $AB$, $F$ is on $AC$, and $AE = 2AF$. Let $G$ be the intersection of $EF$ and $AM$. Compute $\frac{EG}{GF}$ . [u]Round 7[/u] [b]p19.[/b] Find the remainder when $2019x^{2019} -2018x^{2018}+ 2017x^{2017}-...+x$ is divided by $x +1$. [b]p20.[/b] Parallelogram $ABCD$ has $AB = 5$, $BC = 3$, and $\angle ABC = 45^o$. A line through C intersects $AB$ at $M$ and $AD$ at $N$ such that $\vartriangle BCM$ is isosceles. Determine the maximum possible area of $\vartriangle MAN$. [b]p21[/b]. Determine the number of convex hexagons whose sides only lie along the grid shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/9/93cf897a321dfda282a14e8f1c78d32fafb58d.png[/img] [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 4$, $BC = 5$, and $C A = 6$. Extend ray $\overrightarrow{AB}$ to a point $D$ such that $AD = 12$, and similarly extend ray $\overrightarrow{CB}$ to point $E$ such that $CE = 15$. Let $M$ and $N$ be points on the circumcircles of $ABC$ and $DBE$, respectively, such that line $MN$ is tangent to both circles. Determine the length of $MN$. [b]p23.[/b] A volcano will erupt with probability $\frac{1}{20-n}$ if it has not erupted in the last $n$ years. Determine the expected number of years between consecutive eruptions. [b]p24.[/b] If $x$ and $y$ are integers such that $x+ y = 9$ and $3x^2+4x y = 128$, find $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165997p28809441]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $g(x)=1+2x+3x^2+4x^3+\ldots$. Find the coefficient of $x^{2015}$ of $f(x)=\frac{g(x)}{1-x}$.

For any positive integers $n$, find all $n$-tuples of complex numbers $(a_1,..., a_n)$ satisfying $$(x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n.$$ Proposed by USJL.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

Let $f :\mathbb{N} \rightarrow\mathbb{N}$ be an injective function such that $f(1) = 2,~ f(2) = 4$ and \[f(f(m) + f(n)) = f(f(m)) + f(n)\] for all $m, n \in \mathbb{N}$. Prove that $f(n) = n + 2$ for all $n \ge 2$.

Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that $$\displaystyle P(x)=R(T(x))$$