Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$ If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

Solve $(x-4)(x-5)(x-6)(x-7)=1680$

(a) For non negetive $a,b,c, r$ prove that \[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \] (b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side. (c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$

Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

Given $n \geq 2$ distinct integers, which are bigger than $-10$. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even to the biggest of odd. a) Find the smallest $n$ possible b) Find the biggest $n$ possible

Find all solutions of the system consisting of $3$ equations: $x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.

(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \] (b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions: (a) $x_0$ is an even integer; (b) $|x_0|<1000$.

Solve the equation $ 2(x^2\minus{}3x\plus{}2)\equal{}3 \sqrt{x^3\plus{}8}$, where $ x\in R$

Find all polynomilals $P$ with real coefficients, such that $(x+1)P(x-1)+(x-1)P(x+1)=2xP(x)$

Let $a,b,c>1$. Solve in $\mathbb{R}$ the equation $\log_{a+b}(a^x+b)=\log_b((b+c)^x-c)$. [i]Mihai Opincariu[/i]

A positive interger $n$ is called [i][u]rising[/u][/i] if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called [i][u]interger-valued[/u][/i] if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called [i][u]rising-valued[/u][/i] if for all [i]rising[/i] numbers $n$, $P(n)$ takes integer values. Does it necessarily mean that, "every [i]rising-valued[/i] $P$ is also [i]interger-valued[/i] $P$"?

Let the maximum and minimum value of $f(x)=\cos \left(x^{2018}\right) \sin x$ are $M$ and $m$ respectively where $x \in[-2 \pi, 2 \pi] .$ Then $$ M+m= $$ [list=1] [*] $\frac{1}{2}$ [*] $-\frac{1}{\sqrt{2}}$ [*] $\frac{1}{2018}$ [*] Does not exists [/list]

There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5: 4: 3,$ you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?

Let $b \geq 2$ be a positive integer. (a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits. (b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits. (c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even. (d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.

Find out which of the two numbers is greater: $$\dfrac{2}{2 +\dfrac{2}{2 +\dfrac{2}{... +\dfrac{2}{2+\frac22}}}} \,\,\, \text{or} \,\,\, \dfrac{3}{3 +\dfrac{3}{3 +\dfrac{3}{... +\dfrac{3}{3+\frac33}}}}$$ (Each expression has $2022$ fraction signs.)

[u]Round 1[/u] [b]1.1.[/b] Suppose a certain menu has $3$ sandwiches and $5$ drinks. How many ways are there to pick a meal so that you have exactly a drink and a sandwich? [b]1.2.[/b] If $a + b = 4$ and $a + 3b = 222222$, find $10a + b$. [b]1.3.[/b] Compute $$\left\lfloor \frac{2019 \cdot 2017}{2018} \right\rfloor $$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. [u]Round 2[/u] [b]2.1.[/b] Andrew has $10$ water bottles, each of which can hold at most $10$ cups of water. Three bottles are thirty percent filled, five are twenty-four percent filled, and the rest are empty. What is the average amount of water, in cups, contained in the ten water bottles? [b]2.2.[/b] How many positive integers divide $195$ evenly? [b]2.3.[/b] Square $A$ has side length $\ell$ and area $128$. Square $B$ has side length $\ell/2$. Find the length of the diagonal of Square $B$. [u]Round 3[/u] [b]3.1.[/b] A right triangle with area $96$ is inscribed in a circle. If all the side lengths are positive integers, what is the area of the circle? Express your answer in terms of $\pi$. [b]3.2.[/b] A circular spinner has four regions labeled $3, 5, 6, 10$. The region labeled $3$ is $1/3$ of the spinner, $5$ is $1/6$ of the spinner, $6$ is $1/10$ of the spinner, and the region labeled $10$ is $2/5$ of the spinner. If the spinner is spun once randomly, what is the expected value of the number on which it lands? [b]3.3.[/b] Find the integer k such that $k^3 = 8353070389$ [u]Round 4[/u] [b]4.1.[/b] How many ways are there to arrange the letters in the word [b]zugzwang [/b] such that the two z’s are not consecutive? [b]4.2.[/b] If $O$ is the circumcenter of $\vartriangle ABC$, $AD$ is the altitude from $A$ to $BC$, $\angle CAB = 66^o$ and $\angle ABC = 44^o$, then what is the measure of $\angle OAD$ ? [b]4.3.[/b] If $x > 0$ satisfies $x^3 +\frac{1}{x^3} = 18$, find $x^5 +\frac{1}{x^5}$ [u]Round 5[/u] [b]5.1.[/b] Let $C$ be the answer to Question $3$. Neethen decides to run for school president! To be entered onto the ballot, however, Neethen needs $C + 1$ signatures. Since no one else will support him, Neethen gets the remaining $C$ other signatures through bribery. The situation can be modeled by $k \cdot N = 495$, where $k$ is the number of dollars he gives each person, and $N$ is the number of signatures he will get. How many dollars does Neethen have to bribe each person with to get exactly C signatures? [b]5.2.[/b] Let $A$ be the answer to Question $1$. With $3A - 1$ total votes, Neethen still comes short in the election, losing to Serena by just $1$ vote. Darn! Neethen sneaks into the ballot room, knowing that if he destroys just two ballots that voted for Serena, he will win the election. How many ways can Neethen choose two ballots to destroy? [b]5.3.[/b] Let $B$ be the answer to Question $2$. Oh no! Neethen is caught rigging the election by the principal! For his punishment, Neethen needs to run the perimeter of his school three times. The school is modeled by a square of side length $k$ furlongs, where $k$ is an integer. If Neethen runs $B$ feet in total, what is $k + 1$? (Note: one furlong is $1/8$ of a mile). [u]Round 6[/u] [b]6.1.[/b] Find the unique real positive solution to the equation $x =\sqrt{6 + 2\sqrt6 + 2x}- \sqrt{6 - 2\sqrt6 - 2x} -\sqrt6$. [b]6.2.[/b] Consider triangle ABC with $AB = 13$ and $AC = 14$. Point $D$ lies on $BC$, and the lengths of the perpendiculars from $D$ to $AB$ and $AC$ are both $\frac{56}{9}$. Find the largest possible length of $BD$. [b]6.3.[/b] Let $f(x, y) = \frac{m}{n}$, where $m$ is the smallest positive integer such that $x$ and $y$ divide $m$, and $n$ is the largest positive integer such that $n$ divides both $x$ and $y$. If $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, what is the median of the distinct values that $f(a, b)$ can take, where $a, b \in S$? [u]Round 7[/u] [b]7.1.[/b] The polynomial $y = x^4 - 22x^2 - 48x - 23$ can be written in the form $$y = (x - \sqrt{a} - \sqrt{b} - \sqrt{c})(x - \sqrt{a} +\sqrt{b} +\sqrt{c})(x +\sqrt{a} -\sqrt{b} +\sqrt{c})(x +\sqrt{a} +\sqrt{b} -\sqrt{c})$$ for positive integers $a, b, c$ with $a \le b \le c$. Find $(a + b)\cdot c$. [b]7.2.[/b] Varun is grounded for getting an $F$ in every class. However, because his parents don’t like him, rather than making him stay at home they toss him onto a number line at the number $3$. A wall is placed at $0$ and a door to freedom is placed at $10$. To escape the number line, Varun must reach 10, at which point he walks through the door to freedom. Every $5$ minutes a bell rings, and Varun may walk to a different number, and he may not walk to a different number except when the bell rings. Being an $F$ student, rather than walking straight to the door to freedom, whenever the bell rings Varun just randomly chooses an adjacent integer with equal chance and walks towards it. Whenever he is at $0$ he walks to $ 1$ with a $100$ percent chance. What is the expected number of times Varun will visit $0$ before he escapes through the door to freedom? [b]7.3.[/b] Let $\{a_1, a_2, a_3, a_4, a_5, a_6\}$ be a set of positive integers such that every element divides $36$ under the condition that $a_1 < a_2 <... < a_6$. Find the probability that one of these chosen sets also satisfies the condition that every $a_i| a_j$ if $i|j$. [u]Round 8[/u] [b]8.[/b] How many numbers between $1$ and $100, 000$ can be expressed as the product of at most $3$ distinct primes? Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $AB = 5$, $BC = 4$, and $CA = 3$. Initially, there is an ant at each vertex. The ants start walking at a rate of $1$ unit per second, in the direction $A \to B \to C \to A$ (so the ant starting at $A$ moves along ray $\overrightarrow{AB}$, etc.). For a positive real number $t$ less than$ 3$, let $A(t)$ be the area of the triangle whose vertices are the positions of the ants after $t$ seconds have elapsed. For what positive real number $t$ less than $3$ is $A(t)$ minimized?

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

If $a,b,c,d$ are four positive reals such that $abcd= 1$ , prove that $(1+a) (1+b) (1 +c ) (1 +d ) \geq 16.$

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?