A shepherd left, as an inheritance, to his children a flock of $k$ sheep, distributed as follows: the oldest received $\left\lfloor\frac{k}{2}\right\rfloor$ sheep, the middle one $\left\lfloor\frac{k}{3}\right\rfloor$ sheep and the youngest $\left\lfloor\frac{k}{5}\right\rfloor$ sheep. Knowing that there are no sheep left, determine all possible values for $k$.

A four-digit number $n=\overline{a b c d}$, where $a, b, c$ and $d$ are digits, with $a \neq 0$, is said to be [i]guanaco[/i] if the product $\overline{a b} \times \overline{c d}$ is a positive divisor of $n$. Find all guanaco numbers.

Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is $\tfrac{1}{3}$. The probability that a man has none of the three risk factors given that he does not have risk factor A is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a [i]digital root[/i] of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.

Solve in equation: $ x^2+y^2+z^2+w^2=3(x+y+z+w) $ where $ x,y,z,w $ are positive integers.

We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties: (a) $m$ is the product of two consecutive positive integers; (b) the decimal representation of $m$ consists of two identical blocks with $n$ digits. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 8)[/i]

Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that \[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\] for all $n \ge 1$.

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

a) Find all the integers $m$ and $n$ such that \[9m^2+3n=n^2+8\] b) Let $a,b\in \mathbb{N}^*$ . If $x=a^{a+b}+(a+b)^a$ and $y=a^a+(a+b)^{a+b}$ which one is bigger? [i]Florin Nicoara, Valer Pop[/i]

Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.

Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.

Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. (For example, $2^1$ divides $2150$. It may start with $0$.)

Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.

An infinite set $B$ consisting of positive integers has the following property. For each $a,b \in B$ with $a>b$ the number $\frac{a-b}{(a,b)}$ belongs to $B$. Prove that $B$ contains all positive integers. Here, $(a,b)$ is the greatest common divisor of numbers $a$ and $b$.

Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.

Find all positive integers $d$ that can be written in the form $$ d = \gcd(|x^2 - y| , |y^2 - z| , |z^2 - x|), $$ where $x, y, z$ are pairwise coprime positive integers such that $x^2 \neq y$, $y^2 \neq z$, and $z^2 \neq x$.

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $ \textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad $

[u]Round 1[/u] [b]1.1.[/b] There are $99$ dogs sitting in a long line. Starting with the third dog in the line, if every third dog barks three times, and all the other dogs each bark once, how many barks are there in total? [b]1.2.[/b] Indigo notices that when she uses her lucky pencil, her test scores are always $66 \frac23 \%$ higher than when she uses normal pencils. What percent lower is her test score when using a normal pencil than her test score when using her lucky pencil? [b]1.3.[/b] Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.) [u]Round 2[/u] [b]2.1.[/b] If $x + 3y = 2$, then what is the value of the expression $9^x * 729^y$? [b]2.2.[/b] Lazy Sheep loves sleeping in, but unfortunately, he has school two days a week. If Lazy Sheep wakes up each day before school's starting time with probability $1/8$ independent of previous days, then the probability that Lazy Sheep wakes up late on at least one school day over a given week is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]2.3.[/b] An integer $n$ leaves remainder $1$ when divided by $4$. Find the sum of the possible remainders $n$ leaves when divided by $20$. [u]Round 3[/u] [b]3.1. [/b]Jake has a circular knob with three settings that can freely rotate. Each minute, he rotates the knob $120^o$ clockwise or counterclockwise at random. The probability that the knob is back in its original state after $4$ minutes is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]3.2.[/b] Given that $3$ not necessarily distinct primes $p, q, r$ satisfy $p+6q +2r = 60$, find the sum of all possible values of $p + q + r$. [b]3.3.[/b] Dexter's favorite number is the positive integer $x$, If $15x$ has an even number of proper divisors, what is the smallest possible value of $x$? (Note: A proper divisor of a positive integer is a divisor other than itself.) [u]Round 4[/u] [b]4.1.[/b] Three circles of radius $1$ are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-\sqrt{c}}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relatively prime. Find $a + b + c + d$. [b]4.2. [/b]Evaluate $$\frac{\sqrt{15}}{3} \cdot \frac{\sqrt{35}}{5} \cdot \frac{\sqrt{63}}{7}... \cdot \frac{\sqrt{5475}}{73}$$ [b]4.3.[/b] For any positive integer $n$, let $f(n)$ denote the number of digits in its base $10$ representation, and let $g(n)$ denote the number of digits in its base $4$ representation. For how many $n$ is $g(n)$ an integer multiple of $f(n)$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784571p24468619]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.

We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\] where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.

Let $n\in \mathbb{N}$. Prove that $lcm [1,2,..,n]=lcm [\binom{n}{1},\binom{n}{2},...,\binom{n}{n}]$ if and only if $n+1$ is a prime number.

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible.

Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.