Paul starts with the number $19$. In one step, he can add $1$ to his number, divide his number by $2$, or divide his number by $3$. What is the minimum number of steps Paul needs to get to $1$?

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

There are $5$ vertices labelled $1,2,3,4,5$. For any two pairs of vertices $u, v$, the edge $uv$ is drawn with probability $1/2$. If the probability that the resulting graph is a tree is given by $\dfrac{p}{q}$ where $p, q$ are coprime, then find the value of $q^{1/10} + p$.

There are $ n $ natural numbers ($ n\geq 2 $) whose sum is equal to their product. Prove that this common value does not exceed $2n$.

Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n \geq 2$ and sequence of non-integer positive real numbers $a_1, a_2, \dots, a_n$ satisfying $$\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = 1,$$ it's possible to choose positive integers $b_i$ such that (i) for each $i = 1, 2, \dots, n$, either $b_i = \lfloor a_i \rfloor$ or $b_i = \lfloor a_i \rfloor + 1$, and (ii) we have $$1 < \frac{1}{b_1} + \frac{1}{b_2} + \cdots + \frac{1}{b_n} \leq C.$$ (Here $\lfloor \bullet \rfloor$ denotes the floor function, as usual.) [i]Merlijn Staps[/i]

The functions $f,g:[0,1]\to\mathbb R$ are given with the formulas $$f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),$$ and $c$ denotes any solution of $x=f(x)$. (a) i. Analyze the function $f(x)$ and draw its graph. Prove that the equation $f(x)=x$ has the unique root $c$ satisfying $c\in[0.72,0.73]$. ii. Analyze the function $f'(x)$. Let $M_1$ and $M_2$ be the points of the graph of $f(x)$ with different $x$ coordinates. What is the position of the arc $M_1M_2$ of the graph with respect to the segment $M_1M_2$? iii. Analyze the function $g(x)$ and draw its graph. What is the position of that graph with respect to the line $y=x$? Find the tangents to the graph at points with $x$ coordinates $0$ and $1$. iv. Prove that every sequence $\{a_n\}$ with the conditions $a_1\in(0,1)$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$ converges. [hide=Official Hint]Consider the sequences $\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)$ and the function $g(x)$ associated with the graph.[/hide] (b) On the graph of the function $f(x)$ consider the points $M$ and $M'$ with $x$ coordinates $x$ and $f(x)$, where $x\ne c$. i. Prove that the line $MM'$ intersects with the line $y=x$ at the point with $x$ coordinate $$h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.$$ ii. Prove that if $x\in(0,c)$ then $h(x)\in(x,c)$. iii. Analyze whether the sequence $\{a_n\}$ satisfying $a_1\in(0,c),a_{n+1}=h(a_n)$ for $n\in\mathbb N$ converges. Prove that the sequence $\{\tfrac{a_{n+1}-c}{a_n-c}\}$ converges and find its limit. (c) Assume that the calculator approximates every number $b\in[-2,2]$ by number $\overline b$ having $p$ decimal digits after the decimal point. We are performing the following sequence of operations on that calculator: 1) Set $a=0.72$; 2) Calculate $\delta(a)=\overline{f(a)}-a$; 3) If $|\delta(a)|>0.5\cdot10^{-p}$, then calculate $\overline{h(a)}$ and go to the operation $2)$ using $\overline{h(a)}$ instead of $a$; 4) If $|\delta(a)|\le0.5\cdot10^{-p}$, finish the calculation. Let $\overleftrightarrow c$ be the last of calculated values for $\overline{h(a)}$. Assuming that for each $x\in[0.72,0.73]$ we have $\left|\overline{f(x)}-f(x)\right|<\epsilon$, determine $\delta(\overleftrightarrow c)$, the accuracy (depending on $\epsilon$) of the approximation of $c$ with $\overleftrightarrow c$. (d) Assume that the sequence $\{a_n\}$ satisfies $a_1=0.72$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$. Find the smallest $n_0\in\mathbb N$, such that for every $n\ge n_0$ we have $|a_n-c|<10^{-6}$.

Solve the system $\begin{cases} x+ y +z = a \\ x^2 + y^2 + z^2 = a^2 \\ x^3 + y^3 +z^3 = a^3 \end{cases}$

Prove that $$\frac{2}{3}+\frac{4}{5}+\dots +\frac{2010}{2011}$$ is not an integer.

Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?

Given four numbers $x, y, z, t$, let $(a, b, c, d)$ be a permutation of $(x, y, z, t)$ and set $x_1 =|a- b|$, $y_1 = |b-c|$, $z_1 = |c-d|$, and $t_1 = |d -a|$. From $x_1, y_1, z_1, t_1$, form in the same fashion the numbers $x_2, y_2, z_2, t_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z, t_n = t$ for some $n$. Find all possible values of $(x, y, z, t)$.

Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions: (i) $f(f(x))=x^2$ for any $x\geq 1$; (ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$. 1. Prove that $x<f(x)<x^2$ for any $x\geq 1$. 2. Prove that there exists a function $f$ satisfies the above two conditions and the following one: (iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.

Prove that $\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right]$ is a positive integer.

Find the domain of the pairs of positive real numbers $(a,\ b)$ such that there is a $\theta\ (0<\theta \leq \pi)$ such that $\cos a\theta =\cos b\theta$, then draw the domain on the coordinate plane. 30 points

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

Suppose $a_3x^3 - x^2 + a_1x - 7 = 0$ is a cubic polynomial in x whose roots $\alpha,\beta, \gamma$ are positive real numbers satisfying $$\frac{225\alpha^2}{\alpha^2 +7}=\frac{144\beta^2}{\beta^2 +7}=\frac{100\gamma^2}{\gamma^2 +7}.$$ Find $a_1$.

Given integers $a_0,a_1, ... , a_{100}$, satisfying $a_1>a_0$, $a_1>0$, and $a_{r+2}=3 a_{r+1}-2a_r$ for $r=0, 1, ... , 98$. Prove $a_{100}>299$

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]

Number of solutions of the equation $3^x+4^x=8^x$ in reals is [list=1] [*] 0 [*] 1 [*] 2 [*] $\infty$ [/list]

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[f(x+yf(x))+y = xy + f(x+y).\] [i]Proposed by Giannis Galamatis, Greece[/i]

Find all positive integer $n$ such that there exists a $P\in\mathbb{R}[x]$ satisfying $P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}$.

Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

[u]Round 5[/u] [b]p13.[/b] You have a $26 \times 26$ grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of $2 \times 2$ squares that are entirely red? [b]p14.[/b] Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats? [b]p15.[/b] Let $n \ge 1$ be an integer and $a > 0$ a real number. In terms of n, find the number of solutions $(x_1, ..., x_n)$ of the equation $\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2$ such that $x_i$ belongs to the interval $[0, a]$ , for $i = 1, 2, . . . , n$. [u]Round 6 [/u] [b]p16.[/b] All roots of $$\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0$$ are written in the form $r(\cos \phi + i\sin \phi)$ for $i^2 = -1$, $r > 0$, and $0 \le \phi < 2\pi$. What is the smallest positive value of $\phi$ in radians? [b]p17.[/b] Find the sum of the distinct real roots of the equation $$\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.$$ [b]p18.[/b] If $a$ and $b$ satisfy the property that $a2^n + b$ is a square for all positive integers $n$, find all possible value(s) of $a$. [u]Round 7 [/u] [b]p19.[/b] Compute $(1 - \cot 19^o)(1 - \cot 26^o)$. [b]p20.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 5$, and $\angle ABC = 120^o$. Let point $E$ be any point inside $ABC$. The minimum of the sum of the squares of the distances from $E$ to the three sides of $ABC$ can be written in the form $a/b$ , where a and b are natural numbers such that the greatest common divisor of $a$ and $b$ is $1$. Find $a + b$. [b]p21.[/b] Let $m \ne 1$ be a square-free number (an integer – possibly negative – such that no square divides $m$). We denote $Q(\sqrt{m})$ to be the set of all $a + b\sqrt{m}$ where $a$ and $b$ are rational numbers. Now for a fixed $m$, let $S$ be the set of all numbers $x$ in $Q(\sqrt{m})$ such that x is a solution to a polynomial of the form: $x^n + a_1x^{n-1} + .... + a_n = 0$, where $a_0$, $...$, $a_n$ are integers. For many integers m, $S = Z[\frac{m}] = \{a + b\sqrt{m}\}$ where $a$ and $b$ are integers. Give a classification of the integers for which this is not true. (Hint: It is true for $ m = -1$ and $2$.) PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

From the two cities of Dobruisk and Bodruisk, the distance between which is $40$ km, two cyclists Dobi and Bodi simultaneously rode towards each other. Dobie was traveling at $23$ km/h and Bodie was traveling at $17$ km/h. Before departure, a fly landed on Dobie’s nose, which, at the moment of his departure from the city, flew towards Bodruisk at a speed of $40$ km/h. The fly met Bodie, immediately turned back and flew towards Dobruisk at a speed of $30$ km/h. (The fact is that the wind was blowing from Dobruisk to Bodruisk.) Having met Doby, the fly turned back again, etc. Determine the total path that the fly flew until the moment the cyclists met. (The speed of the fly in each direction did not change.)

Given a natural number $n,$ find all polynomials $P(x)$ of degree less than $n$ satisfying the following condition \[ \sum^n_{i=0} P(i) \cdot (-1)^i \cdot \binom{n}{i} = 0. \]