For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$, $$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$ and, for each positive integer $k$ such that $b_{k-1}\geq 2$, $$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$ It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$. Prove that the numbers of terms in the two sequences are always equal.

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

Positive integers $m$ and $n$ have no common divisor greater than one. What is the largest possible value of the greatest common divisor of $m + 2000n$ and $n + 2000m$ ? (S Zlobin)

[b]p1.[/b] Compute $21 \cdot 21 - 20 \cdot 20$. [b]p2.[/b] A square has side length $2$. If the square is scaled by a factor of $n$, the perimeter of the new square is equal to the area of the original square. Find $10n$. [b]p3.[/b] Kevin has $2$ red marbles and $2$ blue marbles in a box. He randomly grabs two marbles. The probability that they are the same color can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p4.[/b] In a classroom, if the teacher splits the students into groups of $3$ or $4$, there is one student left out. If the students formgroups of $5$, every student is in a group. What is the fewest possible number of students in this classroom? [b]p5.[/b] Find the sum of all positive integer values of $x$ such that $\lfloor \sqrt{x!} \rfloor = x$. [b]p6.[/b] Find the number of positive integer factors of $2021^{(2^0+2^1)} \cdot 1202^{(1^2+0^2)}$. [b]p7.[/b] Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not. [b]p8.[/b] In isosceles $\vartriangle ABC$, $AB = AC$, and $\angle ABC = 72^o$. The bisector of $\angle ABC$ intersects $AC$ at $D$. Given that $BC = 30$, find $AD$. [b]p9.[/b] For an arbitrary positive value of $x$, let $h$ be the area of a regular hexagon with side length $x$ and let $s$ be the area of a square with side length $x$. Find the value of $\left \lfloor \frac{10h}{s} \right \rfloor$. [b]p10.[/b] There is a half-full tub of water with a base of $4$ inches by $5$ inches and a height of $8$ inches. When an infinitely long stick with base $1$ inch by $1$ inch is inserted vertically into the bottom of the tub, the number of inches the water level rises by can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p11.[/b] Find the sum of all $4$-digit numbers with digits that are a permutation of the digits in $2021$. Note that positive integers cannot have first digit $0$. [b]p12.[/b] A $10$-digit base $8$ integer is chosen at random. The probability that it has $30$ digits when written in base $2$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p13.[/b] Call a natural number sus if it can be expressed as $k^2 +k +1$ for some positive integer $k$. Find the sum of all sus integers less than $2021$. [b]p14.[/b] In isosceles triangle $ABC$, $D$ is the intersection of $AB$ and the perpendicular to $BC$ through $C$. Given that $CD = 5$ and $AB = BC = 1$, find $\sec^2 \angle ABC$. [b]p15.[/b] Every so often, the minute and hour hands of a clock point in the same direction. The second time this happens after 1:00 is a b minutes later, where a and b are relatively prime positive integers. Find a +b. [b]p16.[/b] The $999$-digit number $N = 123123...123$ is composed of $333$ iterations of the number $123$. Find the least nonnegative integerm such that $N +m$ is a multiple of $101$. [b]p17.[/b] The sum of the reciprocals of the divisors of $2520$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Duncan, Paul, and $6$ Atreides guards are boarding three helicopters. Duncan, Paul, and the guards enter the helicopters at random, with the condition that Duncan and Paul do not enter the same helicopter. Note that not all helicoptersmust be occupied. The probability that Paul has more guards with him in his helicopter than Duncan does can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p19.[/b] Let the minimum possible distance from the origin to the parabola $y = x^2 -2021$ be $d$. The value of d2 can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] In quadrilateral $ABCD$ with interior point $E$ and area $49 \sqrt3$, $\frac{BE}{CE}= 2 \sqrt3$, $\angle ABC = \angle BCD = 90^o$, and $\vartriangle ABC \sim \vartriangle BCD \sim \vartriangle BEC$. The length of $AD$ can be expressed aspn where $n$ is a positive integer. Find $n$. [b]p21.[/b] Find the value of $$\sum^{\infty}_{i=1}\left( \frac{i^2}{2^{i-1}}+\frac{i^2}{2^{i}}+\frac{i^2}{2^{i+1}}\right)=\left( \frac{1^2}{2^{0}}+\frac{1^2}{2^{1}}+\frac{1^2}{2^{2}}\right)+\left( \frac{2^2}{2^{1}}+\frac{2^2}{2^{2}}+\frac{2^2}{2^{3}}\right)+\left( \frac{3^2}{2^{2}}+\frac{2^2}{2^{3}}+\frac{2^2}{2^{4}}\right)+...$$ [b]p22.[/b] Five not necessarily distinct digits are randomly chosen in some order. Let the probability that they form a nondecreasing sequence be $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a +b$ is divided by$ 1000$. [b]p23.[/b] Real numbers $a$, $b$, $c$, and d satisfy $$ac -bd = 33$$ $$ad +bc = 56.$$ Given that $a^2 +b^2 = 5$, find the sum of all possible values of $c^2 +d^2$. [b]p24.[/b] Jeff has a fair tetrahedral die with sides labeled $0$, $1$, $2$, and $3$. He continuously rolls the die and record the numbers rolled in that order. For example, if he rolls a $1$, then rolls a $2$, and then rolls a $3$, he writes down $123$. He keeps rolling the die until he writes the substring $2021$. What is the expected number of times he rolls the die? [b]p25.[/b] In triangle $ABC$, $BC = 2\sqrt3$, and $AB = AC = 4\sqrt3$. Circle $\omega$ with center $O$ is tangent to segment $AB$ at $T$ , and $\omega$ is also tangent to ray $CB$ past $B$ at another point. Points $O, T$ , and $C$ are collinear. Let $r$ be the radius of $\omega$. Given that $r^2 = \frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

All divisors of a positive integer $n$ are listed in the ascending order: $1=d_1<d_2< \ldots < d_k=n$. It turned out that the amount of pairs $(d_i,d_{i+1})$ of adjacent divisors such that $d_{i+1}$ is a multiple of $d_i$, is odd. Prove that $n=pm^2$, where $p$ is the smallest prime divisor of $n$, and $m$ is a positive integer.

In the diagram below $ \angle CAB, \angle CBD$, and $\angle CDE$ are all right angles with side lengths $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$. [asy] size(300); defaultpen(linewidth(0.8)); draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8)); label("$A$",origin,SW); label("$B$",(0,4),dir(160)); label("$C$",(3,0),S); label("$D$",(6,8),dir(80)); label("$E$",(30,-4),E);[/asy]

Let $N>= 2$ be an integer. Show that $4n(N-n)+1$ is never a perfect square for each natural number $n$ less than $N$ if and only if $N^2+1$ is prime.

Find all natural numbers $a>1$, with the property that every prime divisor of $a^6-1$ divides also at least one of the numbers $a^3-1$, $a^2-1$. [i]K. Dochev[/i]

Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.

One day students in school organised a exchange between them such that : $11$ strawberries change for $14$ raspberries, $22$ cherries change for $21$ raspberries, $10$ cherries change for $3$ bananas and $5$ pears for $2$ bananas. How many pears has Amila to give to get $7$ strawberries

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$. $Turkey$

[b]p1.[/b] Solve the system of equations $$xy = 2x + 3y$$ $$yz = 2y + 3z$$ $$zx =2z+3x$$ [b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers. [b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$. [b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor). [b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$. [img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a mathematical competition $n=10\,000$ contestants participate. During the final party, in sequence, the first one takes $1/n$ of the cake, the second one takes $2/n$ of the remaining cake, the third one takes $3/n$ of the cake that remains after the first and the second contestant, and so on until the last one, who takes all of the remaining cake. Determine which competitor takes the largest piece of cake.

Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.

Denote by $\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}$ such that: $(1)$ For all $a,b \in \mathbb{Z}^{*}$ such that $a+b \in \mathbb{Z}^{*}$ we have $f(a+b) \geq $ [b]min[/b] $\left \{ f(a),f(b) \right \}$. $(2)$ For all $a, b \in \mathbb{Z}^{*}$ we have $f(ab) = f(a)+f(b)$.

Show that there are infinitely many triangles with side lengths $a$, $b$, $c$, where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.

Given two integers, $k$ and $d$ such that $d$ divides $k^3 - 2$. Show that there exists integers $a$, $b$, $c$ satisfying $d = a^3 + 2b^3 + 4c^3 - 6abc$. [i]Proposed by Csongor Beke and László Bence Simon, Cambridge[/i]

Given a natural number $n$, for which we can find a prime number less than $\sqrt{n}$ that is not a divisor of $n$. The sequence $a_1, a_2,... ,a_n$ is the numbers $1, 2,... ,n$ arranged in some order. For this sequence, we will find the longest ascending subsequense $a_{i_1} < a_{i_2} < ... < a_{i_k}$, ($i_1 <...< i_k$) and the longest decreasing substring $a_{j_1} > ... > a_{j_l}$, ($j_1 < ... < j_l$) . Prove that at least one of these two subsequnsces $a_{i_1} , . . . , a_{i_k}$ and $a_{j_1} > ... > a_{j_l}$ contains a number that is not a divisor of $n$.

What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ?

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.

Find all functions $f:\mathbb{N}_0 \to \mathbb{Z}$ such that $$f(k)-f(l) \mid k^2-l^2$$ for all integers $k, l \geq 0$.

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.