Pavel and Sara roll two, fair six-sided dice (with faces labeled from $ 1$ to $6$) but do not look at the result. A third-party observer whispers the product of the face-up numbers to Pavel and the sum of the face-up numbers to Sara. Pavel and Sara are perfectly rational and truth-telling, and they both know this. Pavel says, “With the information I have, I am unable to deduce the sum of the two numbers rolled.” Sara responds, “Interesting! With the information I have, I am unable to deduce the product of the two numbers rolled.” Pavel responds, “Wow! I still cannot deduce the sum. But I’m sure you know the product by now!” What is the product?

Find the biggest positive integer $n$ for which the number $(n!)^6-6^n$ is divisible by $2022$.

[i]Round 5[/i] [b]p13.[/b] Kelvin Amphibian, a not-frog who lives on the coordinate plane, likes jumping around. Each step, he jumps either to the spot that is $1$ unit to the right and 2 units up, or the spot that is $2$ units to the right and $1$ unit up, from his current location. He chooses randomly among these two choices with equal probability. He starts at the origin and jumps for a long time. What is the probability that he lands on $(10, 8)$ at some time in his journey? [b]p14.[/b] Points $A, B, C$, and $D$ are randomly chosen on the circumference of a unit circle. What is the probability that line segments $AB$ and $CD$ intersect inside the circle? [b]p15.[/b] Let $P(x)$ be a quadratic polynomial with two consecutive integer roots. If it is also known that $\frac{P(2017)} {P(2016)} = \frac{2016}{2017}$ , find the larger root of $P(x)$. [u]Round 6[/u] [b]p16.[/b] Let $S_n$ be the sum of reciprocals of the integers between $1$ and $n$ inclusive. Find a triple $(a, b, c)$ of positive integers such that $S_{2017} \cdot S_{2017} - S_{2016} \cdot S_{2018} = \frac{S_a+S_b}{c}$ . [b]p17.[/b] Suppose that $m$ and $n$ are both positive integers. Alec has $m$ standard $6$-sided dice, each labelled $1$ to $6$ inclusive on the sides, while James has $n$ standard $12$-sided dice, each labelled $1$ to $12$ inclusive on the sides. They decide to play a game with their dice. They each toss all their dice simultaneously and then compute the sum of the numbers that come up on their dice. Whoever has a higher sum wins (if the sums are equal, they tie). Given that both players have an equal chance of winning, determine the minimum possible value of mn. [b]p18.[/b] Overlapping rectangles $ABCD$ and $BEDF$ are congruent to each other and both have area $1$. Given that $A,C,E, F$ are the vertices of a square, find the area of the square. [u]Round 7[/u] [b]p19.[/b] Find the number of solutions to the equation $$||| ... |||||x| + 1| - 2| + 3| - 4| +... - 98| + 99| - 100| = 0$$ [b]p20.[/b] A split of a positive integer in base $10$ is the separation of the integer into two nonnegative integers, allowing leading zeroes. For example, $2017$ can be split into $2$ and $017$ (or $17$), $20$ and $17$, or $201$ and $7$. A split is called squarish if both integers are nonzero perfect squares. $49$ and $169$ are the two smallest perfect squares that have a squarish split ($4$ and $9$, $16$ and $9$ respectively). Determine all other perfect squares less than $2017$ with at least one squarish split. [b]p21.[/b] Polynomial $f(x) = 2x^3 + 7x^2 - 3x + 5$ has zeroes $a, b$ and $c$. Cubic polynomial $g(x)$ with $x^3$-coefficient $1$ has zeroes $a^2$, $b^2$ and $c2$. Find the sum of coefficients of $g(x)$. [u]Round 8[/u] [b]p22.[/b] Two congruent circles, $\omega_1$ and $\omega_2$, intersect at points $A$ and $B$. The centers of $\omega_1$ and $\omega_2$ are $O_1$ and $O_2$ respectively. The arc $AB$ of $\omega_1$ that lies inside $\omega_2$ is trisected by points $P$ and $Q$, with the points lying in the order $A, P, Q,B$. Similarly, the arc $AB$ of $\omega_2$ that lies inside $\omega_1$ is trisected by points $R$ and $S$, with the points lying in the order $A,R, S,B$. Given that $PQ = 1$ and $PR =\sqrt2$, find the measure of $\angle AO_1B$ in degrees. [b]p23.[/b] How many ordered triples of $(a, b, c)$ of integers between $-10$ and $10$ inclusive satisfy the equation $-abc = (a + b)(b + c)(c + a)$? [b]p24.[/b] For positive integers $n$ and $b$ where $b > 1$, define $s_b(n)$ as the sum of digits in the base-$b$ representation of $n$. A positive integer $p$ is said to dominate another positive integer $q$ if for all positive integers $n$, $s_p(n)$ is greater than or equal to $s_q(n)$. Find the number of ordered pairs $(p, q)$ of distinct positive integers between $2$ and $100$ inclusive such that $p$ dominates $q$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2936487p26278546]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all integers $n\geq 1$ for which the numbers $1,2,\ldots,n$ may be (re)ordered as $a_1,a_2,\ldots,a_n$ in such a way that the average $\dfrac {a_1+a_2+\cdots + a_k} {k}$ is an integer for all values $1\leq k\leq n$. (Dan Schwarz)

[b]p1.[/b] For positive real numbers $x, y$, the operation $\otimes$ is given by $x \otimes y =\sqrt{x^2 - y}$ and the operation $\oplus$ is given by $x \oplus y =\sqrt{x^2 + y}$. Compute $(((5\otimes 4)\oplus 3)\otimes2)\oplus 1$. [b]p2.[/b] Janabel is cutting up a pizza for a party. She knows there will either be $4$, $5$, or $6$ people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices? [b]p3.[/b] If the numerator of a certain fraction is added to the numerator and the denominator, the result is $\frac{20}{19}$ . What is the fraction? [b]p4.[/b] Let trapezoid $ABCD$ be such that $AB \parallel CD$. Additionally, $AC = AD = 5$, $CD = 6$, and $AB = 3$. Find $BC$. [b]p5.[/b] AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream? [b]p6.[/b] Find the minimum possible value of the expression $|x+1|+|x-4|+|x-6|$. [b]p7.[/b] How many $3$ digit numbers have an even number of even digits? [b]p8.[/b] Given that the number $1a99b67$ is divisible by $7$, $9$, and $11$, what are $a$ and $b$? Express your answer as an ordered pair. [b]p9.[/b] Let $O$ be the center of a quarter circle with radius $1$ and arc $AB$ be the quarter of the circle’s circumference. Let $M$,$N$ be the midpoints of $AO$ and $BO$, respectively. Let $X$ be the intersection of $AN$ and $BM$. Find the area of the region enclosed by arc $AB$, $AX$,$BX$. [b]p10.[/b] Each square of a $5$-by-$1$ grid of squares is labeled with a digit between $0$ and $9$, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by $3$. How many such labelings are possible if each digit can be used more than once? [b]p11.[/b] A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is $5$, how many different possible values of the units digit are there? [b]p12.[/b] There are $2019$ red balls and $2019$ white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red? [b]p13.[/b] Let $ABCD$ be a square with side length $2$. Let $\ell$ denote the line perpendicular to diagonal $AC$ through point $C$, and let $E$ and $F$ be themidpoints of segments $BC$ and $CD$, respectively. Let lines $AE$ and $AF$ meet $\ell$ at points $X$ and $Y$ , respectively. Compute the area of $\vartriangle AXY$ . [b]p14.[/b] Express $\sqrt{21-6\sqrt6}+\sqrt{21+6\sqrt6}$ in simplest radical form. [b]p15.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length two. Let $D$ and $E$ be on $AB$ and $AC$ respectively such that $\angle ABE =\angle ACD = 15^o$. Find the length of $DE$. [b]p16.[/b] $2018$ ants walk on a line that is $1$ inch long. At integer time $t$ seconds, the ant with label $1 \le t \le 2018$ enters on the left side of the line and walks among the line at a speed of $\frac{1}{t}$ inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when $t = 2019$ seconds. [b]p17.[/b] Determine the number of ordered tuples $(a_1,a_2,... ,a_5)$ of positive integers that satisfy $a_1 \le a_2 \le ... \le a_5 \le 5$. [b]p18.[/b] Find the sum of all positive integer values of $k$ for which the equation $$\gcd (n^2 -n -2019,n +1) = k$$ has a positive integer solution for $n$. [b]p19.[/b] Let $a_0 = 2$, $b_0 = 1$, and for $n \ge 0$, let $$a_{n+1} = 2a_n +b_n +1,$$ $$b_{n+1} = a_n +2b_n +1.$$ Find the remainder when $a_{2019}$ is divided by $100$. [b]p20.[/b] In $\vartriangle ABC$, let $AD$ be the angle bisector of $\angle BAC$ such that $D$ is on segment $BC$. Let $T$ be the intersection of ray $\overrightarrow{CB}$ and the line tangent to the circumcircle of $\vartriangle ABC$ at $A$. Given that $BD = 2$ and $TC = 10$, find the length of $AT$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

Determine all triples $(a,b,c)$ of positive integers satisfying the conditions $$\gcd(a,20) = b$$ $$\gcd(b,15) = c$$ $$\gcd(a,c) = 5$$ (Richard Henner)

p1. One day, a researcher placed two groups of species that were different, namely amoeba and bacteria in the same medium, each in a certain amount (in unit cells). The researcher observed that on the next day, which is the second day, it turns out that every cell species divide into two cells. On the same day every cell amoeba prey on exactly one bacterial cell. The next observation carried out every day shows the same pattern, that is, each cell species divides into two cells and then each cell amoeba prey on exactly one bacterial cell. Observation on day $100$ shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day. p2. It is known that $n$ is a positive integer. Let $f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$. Find $f(13) + f(14) + f(15) + ...+ f(112).$ p3. Budi arranges fourteen balls, each with a radius of $10$ cm. The first nine balls are placed on the table so that form a square and touch each other. The next four balls placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the four balls. If Bambang has fifty five balls each also has a radius of $10$ cm and all the balls are arranged following the pattern of the arrangement of the balls made by Budi, calculate the height of the center of the topmost ball is measured from the table surface in the arrangement of the balls done by Bambang. p4. Given a triangle $ABC$ whose sides are $5$ cm, $ 8$ cm, and $\sqrt{41}$ cm. Find the maximum possible area of ​​the rectangle can be made in the triangle $ABC$. p5. There are $12$ people waiting in line to buy tickets to a show with the price of one ticket is $5,000.00$ Rp.. Known $5$ of them they only have $10,000$ Rp. in banknotes and the rest is only has a banknote of $5,000.00$ Rp. If the ticket seller initially only has $5,000.00$ Rp., what is the probability that the ticket seller have enough change to serve everyone according to their order in the queue?

For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$. Show that there is no number $n$ with $f (n) = 19091997$.

Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$. [color=blue]It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?[/color]

The numbers $1,2,...,n$ ($n \ge 5$) are written on the circle in the clockwise order. Per move it is allowed to exchange any couple of consecutive numbers $a, b$ to the couple $\frac{a+b}{2}, \frac{a+b}{2}$. Is it possible to make all numbers equal using these operations?

Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$.

How many integers $n$ are there such that $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a prime number?

Find all natural numbers $m$ such that \[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]

Prove that no term of the sequence $10001$, $100010001$, $1000100010001$ , $...$ is prime.

Consider an infinite sequence of positive integers $a_1, a_2, a_3, \dots$ such that $a_1 > 1$ and $(2^{a_n} - 1)a_{n+1}$ is a square for all positive integers $n$. Is it possible for two terms of such a sequence to be equal? [i]Proposed by Pavel Kozlov, Russia[/i]

Find the last four digits of $5^{5555}$.

Let $n \ge 3$ be a positive integer. Suppose that $\Gamma$ is a unit circle passing through a point $A$. A regular $3$-gon, regular $4$-gon, \dots, regular $n$-gon are all inscribed inside $\Gamma$ such that $A$ is a common vertex of all these regular polygons. Let $Q$ be a point on $\Gamma$ such that $Q$ is a vertex of the regular $n$-gon, but $Q$ is not a vertex of any of the other regular polygons. Let $\mathcal{S}_n$ be the set of all such points $Q$. Find the number of integers $3 \le n \le 100$ such that \[ \prod_{Q \in \mathcal{S}_n} |AQ| \le 2. \]

Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and \[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \] for all integers $n \ge 1$. Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$. [i]Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.[/i]

[b]p1.[/b] Suppose $\frac{x}{y} = 0.\overline{ab}$ where $x$ and $y$ are relatively prime positive integers and $ab + a + b + 1$ is a multiple of $12$. Find the sum of all possible values of $y$. [b]p2.[/b] Let $A$ be the set of points $\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}$. How many distinct circles pass through at least three points in $A$? [b]p3.[/b] Jack and Jill need to bring pails of water home. The river is the $x$-axis, Jack is initially at the point $(-5, 3)$, Jill is initially at the point $(6, 1)$, and their home is at the point $(0, h)$ where $h > 0$. If they take the shortest paths home given that each of them must make a stop at the river, they walk exactly the same total distance. What is $h$? [b]p4.[/b] What is the largest perfect square which is not a multiple of $10$ and which remains a perfect square if the ones and tens digits are replaced with zeroes? [b]p5.[/b] In convex polygon $P$, each internal angle measure (in degrees) is a distinct integer. What is the maximum possible number of sides $P$ could have? [b]p6.[/b] How many polynomials $p(x)$ of degree exactly $3$ with real coefficients satisfy $$p(0), p(1), p(2), p(3) \in \{0, 1, 2\}?$$ [b]p7.[/b] Six spheres, each with radius $4$, are resting on the ground. Their centers form a regular hexagon, and adjacent spheres are tangent. A seventh sphere, with radius $13$, rests on top of and is tangent to all six of these spheres. How high above the ground is the center of the seventh sphere? [b]p8.[/b] You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules? [b]p9.[/b] Quadrilateral $ABCD$ has $\overline{AB} = 40$, $\overline{CD} = 10$, $\overline{AD} = \overline{BC}$, $m\angle BAD = 20^o$, and $m \angle ABC = 70^o$. What is the area of quadrilateral $ABCD$? [b]p10.[/b] We say that a permutation $\sigma$ of the set $\{1, 2,..., n\}$ preserves divisibilty if $\sigma (a)$ divides $\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\{1, 2,..., 40\}$ preserve divisibility? (A permutation of $\{1, 2,..., n\}$ is a function $\sigma$ from $\{1, 2,..., n\}$ to itself such that for any $b \in \{1, 2,..., n\}$, there exists some $a \in \{1, 2,..., n\}$ satisfying $\sigma (a) = b$.) [b]p11.[/b] In the diagram shown at right, how many ways are there to remove at least one edge so that some circle with an “A” and some circle with a “B” remain connected? [img]https://cdn.artofproblemsolving.com/attachments/8/7/fde209c63cc23f6d3482009cc6016c7cefc868.png[/img] [b]p12.[/b] Let $S$ be the set of the $125$ points in three-dimension space of the form $(x, y, z)$ where $x$, $y$, and $z$ are integers between $1$ and $5$, inclusive. A family of snakes lives at the point $(1, 1, 1)$, and one day they decide to move to the point $(5, 5, 5)$. Snakes may slither only in increments of $(1,0,0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Given that at least one snake has slithered through each point of $S$ by the time the entire family has reached $(5, 5, 5)$, what is the smallest number of snakes that could be in the family? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square. [i]Proposed by Ilir Snopce[/i]

Prove that no number of the form $10^{-n}$, $n\geq 1,$ can be represented as the sum of reciprocals of factorials of different positive integers.

Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct

Given \(n \in \mathbb{N}\), let \(\sigma (n)\) denote the sum of the divisors of \(n\) and \(\phi (n)\) denote the number of integers \(n \geq m\) for which \(\gcd(m,n) = 1\). Show that for all \(n \in \mathbb{N}\), \[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\] and determine when equality holds.

Solve in integers the equation $x^6+x^5+4=y^2. $ [i]Ioan Cucurezeanu[/i]