Find all positive integers $n$ for which there are distinct integers $a_1$, ..., $a_n$ such that $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1+a_2+...+a_n}{2}$.

A positive integer has exactly $50$ divisors. Is it possible that no difference of two different divisors is divisible by $100$? [i]Proposed by A. Chironov[/i]

A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

Show that, if $n \neq 2$ is a positive integer, that there are $n$ triangular numbers $a_1$, $a_2$, $\ldots$, $a_n$ such that $\displaystyle{\sum_{i=1}^n \frac1{a_i} = 1}$ (Recall that the $k^{th}$ triangular number is $\frac{k(k+1)}2$).

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.

Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.

The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.

Show that there do not exist five consecutive integers whose sum of squares is itself a perfect square.

For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times. Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.

Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.

A natural number $n$ is called [i]nice[/i] if it enjoys the following properties: • The expression is made ​​up of $4$ decimal digits; • the first and third digits of $n$ are equal; • the second and fourth digits of $n$ are equal; • the product of the digits of $n$ divides $n^2$. Determine all nice numbers.

Determine if there exists a positive integer $n$ such that $n^2+11$ is a prime number and $n+4$ is a perfect cube.

Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.

Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$. [i]Ciprian Manolescu[/i]

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of positive integers $m$ between $1$ and $2022$ inclusive such that \[ \left\lfloor \frac{3^m}{11} \right\rfloor \] is even.

Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.

Can it happen that the sum of digits of some positive integer $n$ equals $100$ while the sum of digits of number $n^3$ equals $100^3$?

When Andris entered the room, there were the numbers $3$ and $24$ on the board. In one step, if there are the (not necessarily different) numbers $k$ and $n$ on the board already, then Andris can write the number$ kn + k + n$ on the board, too. a) Can Andris write the number $9999999$ on the board after a few moves? b) What if he wants to get $99999999$? c) And what about $48999999$?

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]

Find all prime numbers $p$ and $q$ such that $3p^4+5q^4+15=13p^2q^2$.

[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].

If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

Let $x,y$ be integer number with $x,y\neq-1$ so that $\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}$. Prove that $x^{4}y^{44}-1$ is divisble by $x+1$