[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent: $\quad(\text{i})\: n|a^n-a$ for any positive integer $a$; $\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.

It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.

Given positive integer $k$, prove that there exists a positive integer $N$ depending only on $k$ such that for any integer $n\geq N$, $\binom{n}{k}$ has at least $k$ different prime divisors.

Find all integers m, n such that $m^3 = n^3 + n$.

Let n be an odd positive integer.Let M be a set of $n$ positive integers ,which are 2x2 different. A set $T$ $\in$ $M$ is called "good" if the product of its elements is divisible by the sum of the elements in $M$, but is not divisible by the square of the same sum. Given that $M$ is "good",how many "good" subsets of $M$ can there be?

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying \[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \] for all positive integer $m,n$.

The Fibonacci sequence is defined by $F_0=F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$. Prove that the sum of $2000$ consecutive terms of the Fibonacci sequence is never a term of the sequence.

[u]Round 9[/u] [b]p25.[/b] Circle $\omega_1$ has radius $1$ and diameter $AB$. Let circle $\omega_2$ be a circle withm aximum radius such that it is tangent to $AB$ and internally tangent to $\omega_1$. A point $C$ is then chosen such that $\omega_2$ is the incircle of triangle $ABC$. Compute the area of $ABC$. [b]p26.[/b] Two particles lie at the origin of a Cartesian plane. Every second, the first particle moves from its initial position $(x, y)$ to one of either $(x +1, y +2)$ or $(x -1, y -2)$, each with probability $0.5$. Likewise, every second the second particle moves from it’s initial position $(x, y)$ to one of either $(x +2, y +3)$ or $(x -2, y -3)$, each with probability $0.5$. Let $d$ be the distance distance between the two particles after exactly one minute has elapsed. Find the expected value of $d^2$. [b]p27.[/b] Find the largest possible positive integer $n$ such that for all positive integers $k$ with $gcd (k,n) = 1$, $k^2 -1$ is a multiple of $n$. [u]Round 10[/u] [b]p28.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $C A = 15$. Let $H$ be the orthcenter of $\vartriangle ABC$, $M$ be the midpoint of segment $BC$, and $F$ be the foot of altitude from $C$ to $AB$. Let $K$ be the point on line $BC$ such that $\angle MHK = 90^o$. Let $P$ be the intersection of $HK$ and $AB$. Let $Q$ be the intersection of circumcircle of $\vartriangle FPK$ and $BC$. Find the length of $QK$. [b]p29.[/b] Real numbers $(x, y, z)$ are chosen uniformly at random from the interval $[0,2\pi]$. Find the probability that $$\cos (x) \cdot \cos (y)+ \cos(y) \cdot \cos (z)+ \cos (z) \cdot \cos(x) + \sin (x) \cdot \sin (y)+ \sin (y) \cdot \sin (z)+ \sin (z) \cdot \sin (x)+1$$ is positive. [b]p30.[/b] Find the number of positive integers where each digit is either $1$, $3$, or $4$, and the sum of the digits is $22$. [u]Round 11[/u] [b]p31.[/b] In $\vartriangle ABC$, let $D$ be the point on ray $\overrightarrow{CB}$ such that $AB = BD$ and let $E$ be the point on ray $\overrightarrow{AC}$ such that $BC =CE$. Let $L$ be the intersection of $AD$ and circumcircle of $\vartriangle ABC$. The exterior angle bisector of $\angle C$ intersects $AD$ at $K$ and it follows that $AK = AB +BC +C A$. Given that points $B$, $E$, and $L$ are collinear, find $\angle C AB$. [b]p32.[/b] Let $a$ be the largest root of the equation $x^3 -3x^2 +1 0$. Find the remainder when $\lfloor a^{2019} \rfloor$ is divided by $17$. [b]p33.[/b] For all $x, y \in Q$, functions $f , g ,h : Q \to Q$ satisfy $f (x + g (y)) = g (h( f (x)))+ y$. If $f (6)=2$, $g\left( \frac12 \right) = 2$, and $h \left( \frac72 \right)= 2$, find all possible values of $f (2019)$. [u]Round 12[/u] [b]p34.[/b] An $n$-polyomino is formed by joining $n$ unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections that turns one into the other. Let $P(n)$ be the number of free $n$-polyominos. For example, $P(3) = 2$ and $P(4) = 5$. Estimate $P(20)+P(19)$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ [b]p35.[/b] Estimate $$\sum^{2019}_{k=1} sin(k),$$ where $k$ is measured in radians. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $\max \, (0,15-10 \cdot |E - A|)$ . [b]p36.[/b] For a positive integer $n$, let $r_{10}(n)$ be the number of $10$-tuples of (not necessarily positive) integers $(a_1,a_2,... ,a_9,a_{10})$ such that $$a^2_1 +a^2_2+ ...+a^2_9+a^2_{10}= n.$$ Estimate $r_{10}(20)+r_{10}(19)$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be$$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165997p28809441]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all triples $(x,y,p)$ of positive integers such that $p$ is prime, $p=x^2+1$ and $2p^2=y^2+1$.

Bob writes $36$ consecutive positive integers in a white paper(in ascending order), next he computes the sum of digits of each one of $36$ numbers(in the order) and writes the first $16$ results in a red paper and the last $10$ results in a blue paper. Determine if Bob can choose the $36$ integers, such that the sum of the numbers in the red paper is less than or equal to sum of the numbers in the blue paper.

For every positive integer \( n \), do there exist pairwise distinct positive integers \( a_1, a_2, \dots, a_n \) that satisfy the following condition? For every \( 3 \leq m \leq n \), there exists an \( i \leq m-2 \) such that: $$ a_m = a_{\gcd(m-1, i)} + \gcd(a_{m-1}, a_i). $$ Proposed by Alireza Jannati

Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.

There exists infinitely many positive integers such that $a^3 + 1990b^3 = c^4$.

Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$

On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.

For $\forall$ $m\in \mathbb{N}$ with $\pi (m)$ we denote the number of prime numbers that are no bigger than $m$. Find all pairs of natural numbers $(a,b)$ for which there exist polynomials $P,Q\in \mathbb{Z}[x]$ so that for $\forall$ $n\in \mathbb{N}$ the following equation is true: $\frac{\pi (an)}{\pi (bn)} =\frac{P(n)}{Q(n)}$.

In a french village the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people would be $ 1$ bigger than a perfect square. If again $100$ more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?

Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$. [i]Proposed by Ankan Bhattacharya[/i]