Solve the equation in positive integers: $2^x-3^y 5^z=1009$.

For a positive integer $n$, let $f(n)$ be the number of factors of $n^2+n+1$. Show that there are infinitely many integers $n$ which satisfy $f(n) \geq f(n+1)$.

Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$

Given are $n$ integers, not necessarily distinct, and two positive integers $p$ and $q$. If the $n$ numbers are not all distinct, choose two equal ones. Add $p$ to one of them and subtract $q$ from the other. If there are still equal ones among the $n$ numbers, repeat this procedure. Prove that after a finite number of steps, all $n$ numbers are distinct.

Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)

[b]p1.[/b] In rectangle $ABMC$, $AB= 5$ and $BM= 8$. If point $X$ is the midpoint of side $AC$, what is the area of triangle $XCM$? [b]p2.[/b] Find the sum of all possible values of $a+b+c+d$ such that $(a, b, c, d)$ are quadruplets of (not necessarily distinct) prime numbers satisfying $a \cdot b \cdot c \cdot d = 4792$. [b]p3.[/b] How many integers from $1$ to $2022$ inclusive are divisible by $6$ or $24$, but not by both? [b]p4.[/b] Jerry begins his English homework at $07:39$ a.m. At $07:44$ a.m., he has finished $2.5\%$ of his homework. Subsequently, for every five minutes that pass, he completes three times as much homework as he did in the previous five minute interval. If Jerry finishes his homework at $AB : CD$ a.m., what is $A + B + C + D$? For example, if he finishes at $03:14$ a.m., $A + B + C + D = 0 + 3 + 1 + 4$. [b]p5.[/b] Advay the frog jumps $10$ times on Mondays, Wednesdays and Fridays. He jumps $7$ times on Tuesdays and Saturdays. He jumps $5$ times on Thursdays and Sundays. How many times in total did Advay jump in November if November $17$th falls on a Thursday? (There are $30$ days in November). [b]p6.[/b] In the following diagram, $\angle BAD\cong \angle DAC$, $\overline{CD} = 2\overline{BD}$, and $ \angle AEC$ and $\angle ACE$ are complementary. Given that $\overline{BA} = 210$ and $\overline{EC} = 525$, find $\overline{AE}$. [img]https://cdn.artofproblemsolving.com/attachments/5/3/8e11caf2d7dbb143a296573f265e696b4ab27e.png[/img] [b]p7.[/b] How many trailing zeros are there when $2021!$ is expressed in base $2021$? [b]p8.[/b] When two circular rings of diameter $12$ on the Olympic Games Logo intersect, they meet at two points, creating a $60^o$ arc on each circle. If four such intersections exist on the logo, and no region is in $3$ circles, the area of the regions of the logo that exist in exactly two circles is $a\pi - b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $\sqrt{c}$ is fully simplified find $a + b + c$. [b]p9.[/b] If $x^2 + ax - 3$ is a factor of $x^4 - x^3 + bx^2 - 5x - 3$, then what is $|a + b|$? [b]p10.[/b] Let $(x, y, z)$ be the point on the graph of $x^4 +2x^2y^2 +y^4 -2x^2 -2y^2 +z^2 +1 = 0$ such that $x+y +z$ is maximized. Find $a+b$ if $xy +xz +yz$ can be expressed as $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers. [b]p11.[/b] Andy starts driving from Pittsburgh to Columbus and back at a random time from $12$ pm to $3$ pm. Brendan starts driving from Pittsburgh to Columbus and back at a random time from $1$ pm to $4$ pm. Both Andy and Brendan take $3$ hours for the round trip, and they travel at constant speeds. The probability that they pass each other closer to Pittsburgh than Columbus is$ m/n$, for relatively prime positive integers $m$ and $n$. What is $m + n$? [b]p12.[/b] Consider trapezoid $ABCD$ with $AB$ parallel to $CD$ and $AB < CD$. Let $AD \cap BC = O$, $BO = 5$, and $BC = 11$. Drop perpendicular $AH$ and $BI$ onto $CD$. Given that $AH : AD = \frac23$ and $BI : BC = \frac56$ , calculate $a + b + c + d - e$ if $AB + CD$ can be expressed as $\frac{a\sqrt{b} + c\sqrt{d}}{e}$ where $a$, $b$, $c$, $d$, $e$ are integers with $gcd(a, c, e) = 1$ and $\sqrt{b}$, $\sqrt{d}$ are fully simplified. [b]p13.[/b] The polynomials $p(x)$ and $q(x)$ are of the same degree and have the same set of integer coefficients but the order of the coefficients is different. What is the smallest possible positive difference between $p(2021)$ and $q(2021)$? [b]p14.[/b] Let $ABCD$ be a square with side length $12$, and $P$ be a point inside $ABCD$. Let line $AP$ intersect $DC$ at $E$. Let line $DE$ intersect the circumcircle of $ADP$ at $F \ne D$. Given that line $EB$ is tangent to the circumcircle of $ABP$ at $B$, and $FD = 8$, find $m + n$ if $AP$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. [b]p15.[/b] A three digit number $m$ is chosen such that its hundreds digit is the sum of the tens and units digits. What is the smallest positive integer $n$ such that $n$ cannot divide $m$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there are 100 natural number $a_1 < a_2 < ... < a_{99} < a_{100}$ ( $ a_i < 10^6$) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ? $A = \{a_1 , a_2 ,... , a_{99} ,a_{100}\}$ $2A = \{2a_i \vert 1\leq i\leq 100\}$ $A+A = \{a_i + a_j \vert 1\leq i<j\leq 100\}$ $A + 2A = \{a_i + 2a_j \vert 1\leq i,j\leq 100\}$ $2A + 2A = \{2a_i + 2a_j \vert 1\leq i<j\leq 100\}$ (20 ponits )

Given natural $n$. Consider all the fractions $1/(pq)$, where $p$ and $q$ are relatively prime, $0<p<q\le n , p+q>n$. Prove that the sum of all such a fractions equals to $1/2$.

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.

Show that for any rational number $a$ the equation $y =\sqrt{x^2 +a}$ has infinitely many solutions in rational numbers $x$ and $y$.

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?

Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation \[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

Does there exist on the Cartesian plane a convex $2023$-gon with vertices at integer points, such that the lengths of all its sides are equal? [i]Proposed by Anton Trygub[/i]

Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.

Petya wrote down 100 positive integers $n, n+1, \ldots, n+99$, and Vasya wrote down 99 positive integers $m, m-1, \ldots, m-98$. It turned out that for each of Petya's numbers, there is a number from Vasya that divides it. Prove that $m>n^3/10, 000, 000$. [i]Proposed by Ilya Bogdanov[/i]

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

Suppose each element $i\in S=\{1,2,\ldots,n\}$ is assigned a nonempty set $S_i\subseteq S$ so that the following conditions are fulfilled: (i) for any $i,j\in S$, if $j\in S_i$ then $i\in S_j$; (ii) for any $i,j\in S$, if $|S_i|=|S_j|$ then $S_i\cap S_j=\emptyset$. Prove that there exists $k\in S$ for which $|S_k|=1$.

For every $n \in \mathbb{N}$, there exist integers $k$ such that $n | k$ and $k$ contains only zeroes and ones in its decimal representation. Let $f(n)$ denote the least possible number of ones in any such $k$. Determine whether there exists a constant $C$ such that $f(n) < C$ for all $n \in \mathbb{N}$.

Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.

Let $p>10$ be a prime. Prove that there is positive integers $m,n$ with $m+n<p$ such that $p$ divides $5^m7^n -1$

Let $F_k$ denote the series of Fibonacci numbers shifted back by one index, so that $F_0 = 1$, $F_1 = 1,$ and $F_{k+1} = F_k +F_{k-1}$. It is known that for any fixed $n \ge 1$ there exist real constants $b_n$, $c_n$ such that the following recurrence holds for all $k \ge 1$: $$F_{n\cdot (k+1)} = b_n \cdot F_{n \cdot k} + c_n \cdot F_{n\cdot (k-1)}.$$ Prove that $|c_n| = 1$ for all $n \ge 1$.

Two players take turns writing on the board in a row from left to right arbitrary numbers. The player loses, after whose move one or more several digits written in a row form a number divisible by $11$. Which player will win if played correctly?

Determine all positive integers $n$ less than $2024$ such that for all positive integers $x$, the greatest common divisor of $9x + 1$ and $nx+1$ is $1$.