[u]Set 4[/u] [b]p10.[/b] Eve has nine letter tiles: three $C$’s, three $M$’s, and three $W$’s. If she arranges them in a random order, what is the probability that the string “$CMWMC$” appears somewhere in the arrangement? [b]p11.[/b] Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$4$ per package, and $D$ batteries for $\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$2021$, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell? [b]p12.[/b] A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck. PS. You should use hide for answers.

Let $x$ be the first term in the sequence $31, 331, 3331, . . .$ which is divisible by $17$. How many digits long is$ x$?

Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD\equal{}DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC\equal{}2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.

Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$

Let \(a,b\) be positive integers such that \(a+1\), \(b+1\), and \(ab\) are perfect squares. Prove that $\gcd(a,b)+1$ is also a perfect square.

Let $p\geq 3$ be an odd positive integer. Show that $p$ is prime if and only if however we choose $(p+1)/2$ pairwise distinct positive integers, we can find two of them, $a$ and $b$, such that $(a+b)/\gcd(a,b)\geq p.$

Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.) (A Andjans)

Let $d > 0$ be an arbitrary real number. Consider the set $S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}$. Prove that $S_{n}(d)$ has a maximum element.

Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$. [hide] Stronger one. Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?

$\textbf{N6.}$ Let $a,b$ be positive integers. If $a,b$ satisfy that \begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*} is also a positive integer, show that \begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*} is a Fibonacci number. [i]Proposed by usjl[/i]

For every positive integers $n,m$, show that there exist two sets $A,B$ which satisfy the following. [list] [*]$A$ is a set of $n$ successive positive integers, and $B$ is a set of $m$ successive positive integers. [*]$A\cup B = \phi$ [*]For every $a\in A$ and $b\in B$, $a$ and $b$ are not relatively prime. [/list]

Find the least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ and $b$.

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years). In order to receive credit for the correct answer to a problem, you must also correctly answer $\textit{every one}$ $\textit{of the previous parts}$ $\textit{of the Ultimate Question}$.

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

[b]p1.[/b] Is it possible to write $5$ different fractions that add up to $1$, such that their numerators are equal to one and their denominators are natural numbers? [b]p2.[/b] The following is known about two numbers $x$ and $y$: if $x\ge 0$, then $y = 1 -x$; if $y\le 1$, then $x = 1 + y$; if $x\le 1$, then $x = |1 + y|$. Find $x$ and $y$. [b]p3.[/b] Five people living in different cities received a salary, some more, others less ($143$, $233$, $313$, $410$ and $413$ rubles). Each of them can send money to the other by mail. In this case, the post office takes $10\%$ of the amount of money sent for the transfer (in order to receive $100$ rubles, you need to send $10\%$ more, that is, $110$ rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method? [b]p4.[/b] a) List three different natural numbers $m$, $n$ and $k$ for which $m! = n! \cdot k!$ . b) Is it possible to come up with $1999$ such triplets? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Five distinct $2$-digit numbers are in a geometric progression. Find the middle term.

Let us denote by $<a>$ the distance from $a$ to the nearest integer. (For example, $<1,2> = 0.2$, $<\sqrt3> = 2-\sqrt3$) How many solutions does the system of equations have $$\begin{cases} <19x>=y \\ <96y>=x \end{cases} \,\,\, ?$$

Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$.

Given a prime $p$ in the form $4m+1$ ($m\in\mathbb{Z}$). Prove that the number $216p^3$ can't be represented in the form $x^2+y^2+z^9$, $x,y,z\in\mathbb{Z}$

Let $n>1$ be a positive integer and $d_1<d_2<\dots<d_m$ be its $m$ positive divisors, including $1$ and $n$. Lalo writes the following $2m$ numbers on a board: \[d_1,d_2\dots, d_m,d_1+d_2,d_2+d_3,\dots,d_{m-1}+d_m,N \] where $N$ is a positive integer. Afterwards, Lalo erases any number that is repeated (for example, if a number appears twice, he erases one of them). Finally, Lalo realizes that the numbers left on the board are exactly all the divisors of $N$. Find all possible values that $n$ can take.

Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$.