A sequence of integers $a_1,a_2,\ldots $ is such that $a_1=1,a_2=2$ and for $n\ge 1$, \[a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. \] Prove that $a_n\not= 0$ for all $n$.

For each integer $n \geq 2$, let $F(n)$ denote the greatest prime factor of $n$. A [i]strange pair[/i] is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$. Prove that there exist infinitely many strange pairs.

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

[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].

(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$. (b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]

Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.

Let $d$ denote the greatest common divisor of the natural numbers $a$ and $b$, and let $d'$ denote the greatest common divisor of the natural numbers $a'$ and $b'$. Prove that $dd'$ is the greatest common divisor of the four numbers $$ aa' , \ \ ab' , \ \ ba' , \ \ bb' .$$

Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors.

Let $q=p^r$ for a prime number $p$ and positive integer $r$. Let $\zeta = e^{\frac{2\pi i}{q}}$. Find the least positive integer $n$ such that \[\sum_{\substack{1\leq k\leq q\\ \gcd(k,p)=1}} \frac{1}{(1-\zeta^k)^n}\] is not an integer. (The sum is over all $1\leq k\leq q$ with $p$ not dividing $k$.) [i]Victor Wang[/i]

(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ . (b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.

Find the total number of times the digit ‘$2$’ appears in the set of integers $\{1,2,..,1000\}$. For example, the digit ’$2$’ appears twice in the integer $229$.

Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$.

Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer

Let $a_1, a_2, ... , a_{2012}$ be odd positive integers. Prove that the number $$A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1}$$ is irrational.

Observe that the fraction $ \frac{1}{7}\equal{}0,142857$ is a pure periodical decimal with period $ 6\equal{}7\minus{}1$,and in one period one has $ 142\plus{}857\equal{}999$.For $ n\equal{}1,2,\dots$ find a sufficient and necessary condition that the fraction $ \frac{1}{2n\plus{}1}$ has the same properties as above and find two such fractions other than $ \frac{1}{7}$.

[b]p1.[/b] If $x^2 = 7$, what is $x^4 + x^2 + 1$? [b]p2.[/b] Richard and Alex are competing in a $150$-meter race. If Richard runs at a constant speed of $5$ meters per second and Alex runs at a constant speed of $3$ meters per second, how many more seconds does it take for Alex to finish the race? [b]p3.[/b] David and Emma are playing a game with a chest of $100$ gold coins. They alternate turns, taking one gold coin if the chest has an odd number of gold coins or taking exactly half of the gold coins if the chest has an even number of gold coins. The game ends when there are no more gold coins in the chest. If Emma goes first, how many gold coins does Emma have at the end? [b]p4.[/b] What is the only $3$-digit perfect square whose digits are all different and whose units digit is $5$? [b]p5.[/b] In regular pentagon $ABCDE$, let $F$ be the midpoint of $\overline{AB}$, $G$ be the midpoint of $\overline{CD}$, and $H$ be the midpoint of $\overline{AE}$. What is the measure of $\angle FGH$ in degrees? [b]p6.[/b] Water enters at the left end of a pipe at a rate of $1$ liter per $35$ seconds. Some of the water exits the pipe through a leak in the middle. The rest of the water exits from the right end of the pipe at a rate of $1$ liter per $36$ seconds. How many minutes does it take for the pipe to leak a liter of water? [b]p7.[/b] Carson wants to create a wire frame model of a right rectangular prism with a volume of $2022$ cubic centimeters, where strands of wire form the edges of the prism. He wants to use as much wire as possible. If Carson also wants the length, width, and height in centimeters to be distinct whole numbers, how many centimeters of wire does he need to create the prism? [b]p8.[/b] How many ways are there to fill the unit squares of a $3 \times 5$ grid with the digits $1$, $2$, and $3$ such that every pair of squares that share a side differ by exactly $1$? [b]p9.[/b] In pentagon ABCDE, $AB = 54$, $AE = 45$, $DE = 18$, $\angle A = \angle C = \angle E$, $D$ is on line segment $\overline{BE}$, and line $BD$ bisects angle $\angle ABC$, as shown in the diagram below. What is the perimeter of pentagon $ABCDE$? [img]https://cdn.artofproblemsolving.com/attachments/2/0/7c25837bb10b128a1c7a292f6ce8ce3e64b292.png[/img] [b]p10.[/b] If $x$ and $y$ are nonzero real numbers such that $\frac{7}{x} + \frac{8}{y} = 91$ and $\frac{6}{x} + \frac{10}{y} = 89$, what is the value of $x + y$? [b]p11.[/b] Hilda and Marianne play a game with a shued deck of $10$ cards, numbered from $1$ to $10$. Hilda draws five cards, and Marianne picks up the five remaining cards. Hilda observes that she does not have any pair of consecutive cards - that is, no two cards have numbers that differ by exactly $1$. Additionally, the sum of the numbers on Hilda's cards is $1$ less than the sum of the numbers on Marianne's cards. Marianne has exactly one pair of consecutive cards - what is the sum of this pair? [b]p12.[/b] Regular hexagon $AUSTIN$ has side length $2$. Let $M$ be the midpoint of line segment $\overline{ST}$. What is the area of pentagon $MINUS$? [b]p13.[/b] At a collector's store, plushes are either small or large and cost a positive integer number of dollars. All small plushes cost the same price, and all large plushes cost the same price. Two small plushes cost exactly one dollar less than a large plush. During a shopping trip, Isaac buys some plushes from the store for 59 dollars. What is the smallest number of dollars that the small plush could not possibly cost? [b]p14.[/b] Four fair six-sided dice are rolled. What is the probability that the median of the four outcomes is $5$? [b]p15.[/b] Suppose $x_1, x_2,..., x_{2022}$ is a sequence of real numbers such that: $x_1 + x_2 = 1$ $x_2 + x_3 = 2$ $...$ $x_{2021} + x_{2022} = 2021$ If $x_1 + x_{499} + x_{999} + x_{1501} = 222$, then what is the value of $x_{2022}$? [b]p16.[/b] A cone has radius $3$ and height $4$. An infinite number of spheres are placed in the cone in the following way: sphere $C_0$ is placed inside the cone such that it is tangent to the base of the cone and to the curved surface of the cone at more than one point, and for $i \ge 1$, sphere $C_i$ is placed such that it is externally tangent to sphere $C_{i-1}$ and internally tangent to more than one point of the curved surface of the cone. If $V_i$ is the volume of sphere $C_i$, compute $V_0 + V_1 + V_2 + ... $ . [img]https://cdn.artofproblemsolving.com/attachments/b/4/b43e40bb0a5974dd9d656691c14b4ae268b5b5.png[/img] [b]p17.[/b] Call an ordered pair, $(x, y)$, relatable if $x$ and $y$ are positive integers where $y$ divides $3600$, $x$ divides $y$ and $\frac{y}{x}$ is a prime number. For every relatable ordered pair, Leanne wrote down the positive difference of the two terms of the pair. What is the sum of the numbers she wrote down? [b]p18.[/b] Let $r, s$, and $t$ be the three roots of $P(x) = x^3 - 9x - 9$. Compute the value of $(r^3 + r^2 - 10r - 8)(s^3 + s^2 - 10s - 8)(t^3 + t^2 - 10t - 8)$. [b]p19.[/b] Compute the number of ways to color the digits $0, 1, 2, 3, 4, 5, 6, 7, 8$ and $9$ red, blue, or green such that: (a) every prime integer has at least one digit that is not blue, and (b) every composite integer has at least one digit that is not green. Note that $0$ is not composite. For example, since $12$ is composite, either the digit $1$, the digit $2$, or both must be not green. [b]p20.[/b] Pentagon $ABCDE$ has $AB = DE = 4$ and $BC = CD = 9$ with $\angle ABC = \angle CDE = 90^o$, and there exists a circle tangent to all five sides of the pentagon. What is the length of segment $\overline{AE}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition? [i]We call two integers coprime if they have no common divisor greater than $1$.[/i]

[b]p1.[/b] Minseo writes all of the divisors of $1,000,000$ on the whiteboard. She then erases all of the numbers which have the digit $0$ in their decimal representation. How many numbers are left? [b]p2.[/b] $n < 100$ is an odd integer and can be expressed as $3k - 2$ and $5m + 1$ for positive integers $k$ and $m$. Find the sum of all possible values of $n$. [b]p3.[/b] Mr. Pascal is a math teacher who has the license plate $SQUARE$. However, at night, a naughty student scrambles Mr. Pascal’s license plate to $UQRSEA$. The math teacher luckily has an unscrambler that is able to move license plate letters. The unscrambler swaps the positions of any two adjacent letters. What is the minimum number of times Mr. Pascal must use the unscrambler to restore his original license plate? [b]p4.[/b] Find the number of distinct real numbers $x$ which satisfy $x^2 + 4 \lfloor x \rfloor + 4 = 0$. [b]p5.[/b] All four faces of tetrahedron $ABCD$ are acute. The distances from point $D$ to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ are all $7$, and the distance from point $D$ to face $ABC$ is $5$. Given that the volume of tetrahedron $ABCD$ is $60$, find the surface area of tetrahedron $ABCD$. [b]p6.[/b] Forrest has a rectangular piece of paper with a width of $5$ inches and a height of $3$ inches. He wants to cut the paper into five rectangular pieces, each of which has a width of $1$ inch and a distinct integer height between $1$ and $5$ inches, inclusive. How many ways can he do so? (One possible way is shown below.) [img]https://cdn.artofproblemsolving.com/attachments/7/3/205afe28276f9df1c6bcb45fff6313c6c7250f.png[/img] [b]p7.[/b] In convex quadrilateral $ABCD$, $AB = CD = 5$, $BC = 4$ and $AD = 8$. If diagonal $\overline{AC}$ bisects $\angle DAB$, find the area of quadrilateral $ABCD$. [b]p8.[/b] Let $x$ and $y$ be real numbers such that $$x + y = x^3 + y^3 + \frac34 = \frac{1}{8xy}.$$ Find the value of $x + y$. [b]p9.[/b] Four blue squares and four red parallelograms are joined edge-to-edge alternately to form a ring of quadrilateral as shown. The areas of three of the red parallelograms are shown. Find the area of the fourth red parallelogram. [img]https://cdn.artofproblemsolving.com/attachments/9/c/911a8d53604f639e2f9bd72b59c7f50e43e258.png[/img] [b]p10.[/b] Define $f(x, n) =\sum_{d|n}\frac{x^n-1}{x^d-1}$ . For how many integers $n$ between $1$ and $2023$ inclusive is $f(3, n)$ an odd integer? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve in positive integers the following equation; $xy(x+y-10)-3x^2-2y^2+21x+16y=60$

A wizard thinks of a number from $1$ to $n$. You can ask the wizard any number of yes/no questions about the number. The wizard must answer all those questions, but not necessarily in the respective order. What is the least number of questions that must be asked in order to know what the number is for sure. (In terms of $n$.) Fresh translation.

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

[b]a)[/b] Let $ m,n,p\in\mathbb{Z}_{\ge 0} $ such that $ m>n $ and $ \sqrt{m} -\sqrt n=p. $ Prove that $ m $ and $ n $ are perfect squares. [b]b)[/b] Find the numbers of four digits $ \overline{abcd} $ that satisfy the equation: $$ \sqrt {\overline{abcd} } -\sqrt{\overline{acd}} =\overline{bb} . $$

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]