Which number is greater: $$A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},$$where each of the numbers above contains $1998$ zeros?

Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$

Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.

Let $P(x)=x^4+20x^3+29x^2-666x+2025.$ It is known that $P(x)>0$ for every real $x.$ There is a root $r$ for $P$ in the first quadrant of the complex plane that can be expressed as $r=\frac{1}{2}(a+bi+\sqrt{c+di}),$ where $a,b,c,d$ are integers. Find $a+b+c+d.$

Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.

Bristy wants to build a special set $A$. She starts with $A=\{0, 42\}$. At any step, she can add an integer $x$ to the set $A$ if it is a root of a polynomial which uses the already existing integers in $A$ as coefficients. She keeps doing this, adding more and more numbers to $A$. After she eventually runs out of numbers to add to $A$, how many numbers will be in $A$?

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation $$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$ for any two real numbers $x$ and $y$.

[u]Round 1[/u] [b]p1.[/b] Elaine creates a sequence of positive integers $\{s_n\}$. She starts with $s_1 = 2018$. For $n \ge 2$, she sets $s_n =\frac12 s_{n-1}$ if $s_{n-1}$ is even and $s_n = s_{n-1} + 1$ if $s_{n-1}$ is odd. Find the smallest positive integer $n$ such that $s_n = 1$, or submit “$0$” as your answer if no such $n$ exists. [b]p2.[/b] Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins? [b]p3.[/b] Four circles each of radius $\frac14$ are centered at the points $\left( \pm \frac14, \pm \frac14 \right)$, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle? [u]Round 2 [/u] [b]p4.[/b] If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing? [b]p5.[/b] Let $a_1 = 2018$, and for $n \ge 2$ define $a_n = 2018^{a_{n-1}}$ . What is the ones digit of $a_{2018}$? [b]p6.[/b] We can write $(x + 35)^n =\sum_{i=0}^n c_ix^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$? [u]Round 3[/u] [b]p7.[/b] How many positive integers are factors of $12!$ but not of $(7!)^2$? [b]p8.[/b] How many ordered pairs $(f(x), g(x))$ of polynomials of degree at least $1$ with integer coefficients satisfy $f(x)g(x) = 50x^6 - 3200$? [b]p9.[/b] On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between $1$ and $10$ (inclusive). What is the probability that the average of their three scores is an integer? [u]Round 4[/u] [b]p10.[/b] Find the largest positive integer N such that $$(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)$$ is divisible by $N$ for all choices of positive integers $a > b > c > d > e$. [b]p11.[/b] Let $ABCDE$ be a square pyramid with $ABCD$ a square and E the apex of the pyramid. Each side length of $ABCDE$ is $6$. Let $ABCDD'C'B'A'$ be a cube, where $AA'$, $BB'$, $CC'$, $DD'$ are edges of the cube. Andy the ant is on the surface of $EABCDD'C'B'A'$ at the center of triangle $ABE$ (call this point $G$) and wants to crawl on the surface of the cube to $D'$. What is the length the shortest path from $G$ to $D'$? Write your answer in the form $\sqrt{a + b\sqrt3}$, where $a$ and $b$ are positive integers. [b]p12.[/b] A six-digit palindrome is a positive integer between $100, 000$ and $999, 999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.

Let $\alpha\geq 1$ be a real number. Define the set $$A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}$$ Suppose that all the positive integers that [b]does not belong[/b] to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $2021$ with $0\leq r<2021$. Determine all the possible values of $\alpha$.

One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate $10$ pots of honey and $22$ cups of condensed milk alone, whereby he needed two minutes for each pot of honey and $1$ minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs $5$ minutes for a pot of honey and $3$ minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

Find all real $(x,y)$ such that $x + {y^2} = {y^3}$ $y + {x^2} = {x^3}$

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$; ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$. For each integer $m\ge 2$, find the value of $f(2^m)$.

Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

Two linear functions $f(x)$ and $g(x)$ satisfy the properties that for all $x$, $\bullet$ $f(x) + g(x) = 2$ $\bullet$ $f(f(x)) = g(g(x))$ and $f(0) = 2022$. Compute $f(1)$.

Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying $3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.

Let $a,b,c$ be real numbers. Show that $\sqrt[3]{a} + \sqrt[3]{b} +\sqrt[3]{c} = \sqrt[3]{a+b+c}$ if and only if $ a^3 + b^3 + c^3 = (a + b + c)^3 $

Prove the equality: $$\sum_{k=1}^{n-1}\frac1{\sin^2\frac{(2k+1)\pi}{2n}}=n^2$$ where $n$ is a natural number. [i]H. Lesov[/i]

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)

Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.

Integers $a,b,c$ are such that the values of the trinomials $bx^2+cx+a$ and $cx^2+ax+b$ at $x=1234$ coincide. Can the first trinomial at $x = 1$ take the value $2009$?

Find all functions $f:R \to R$ such that $$f(x^2+f(y)-y) =(f(x))^2-f(y)$$ for all $x,y \in R$