For an integer $n > 0$, let $p(n)$ be the product of the digits of $n$. Compute the sum of all integers $n$ such that $n - p(n) = 52$.

In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.

Let $a$ and $k$ be positive integers such that $a^2+k$ divides $(a-1)a(a+1)$. Prove that $k\ge a$.

Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$?

Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if: (a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$; (b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.

For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) [i]Aprameya Tripathy[/i]

[u]Set 3[/u] [b]p7.[/b] On unit square $ABCD$, a point $P$ is selected on segment $CD$ such that $DP =\frac14$ . The segment $BP$ is drawn and its intersection with diagonal $AC$ is marked as $E$. What is the area of triangle $AEP$? [b]p8.[/b] Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance $1$ apart, two pairs of points are distance $\sqrt3$ apart, and one pair of points is distance $2$ apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be? [b]p9.[/b] The inhabitants of Mars use a base $8$ system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base $8$ numerals $1245415_8$, $7563265_8$, and $ 6321473_8$. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is $1014133027\blacksquare 27662041138$. What is the missing digit? PS. You should use hide for answers.

Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$. [i]Proposed by Mohsen Jamali[/i]

Find all sequences of positive integers $a_1,a_2,\dots$ such that $$(n^2+1)a_n = n(a_{n^2}+1)$$ for all positive integers $n$.

An odd prime $p$ is called a prime of the year $2022$ if there is a positive integer $n$ such that $p^{2022}$ divides $n^{2022}+2022$. Show that there are infinitely many primes of the year $2022$.

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.

Prove that every selection of $1325$ integers from $M=\{1, 2, \cdots, 1987 \}$ must contain some three numbers $\{a, b, c\}$ which are pairwise relatively prime, but that it can be avoided if only $1324$ integers are selected.

Find all pairs of integers $ x $, $ y $ satisfying the equation $$ (xy-1)^2 = (x +1)^2 + (y +1)^2.$$

$8712$ is an integral multiple of its reversal, $2178$, as $8712=4 * 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal.

We call a positive integer a [i]shuffle[/i] number if the following hold: (1) All digits are nonzero. (2) The number is divisible by $11$. (3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$. How many $10$-digit [i]shuffle[/i] numbers are there?

For every integer $n$ not equal to $1$ or $-1$, define $S(n)$ as the smallest integer greater than $1$ that divides $n$. In particular, $S(0)=2$. We also define $S(1) = S(-1) = 1$. Let $f$ be a non-constant polynomial with integer coefficients such that $S(f(n)) \leq S(n)$ for every positive integer $n$. Prove that $f(0)=0$. [b]Note:[/b] A non-constant polynomial with integer coefficients is a function of the form $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_k x^k$, where $k$ is a positive integer and $a_0,a_1,\ldots,a_k$ are integers such that $a_k \neq 0$. [i]Pitchayut Saengrungkongka, Thailand[/i]

Find all positive integers $m$ and $n$ such that both $3^{m}+1$ and $3^{n}+1$ are divisible by $mn$.

Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$. For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$. Show that an odd perfect number has at least $3$ distinct prime divisors. [i]Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.[/i]

Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions. (i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$ (iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$

Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

[u]Round 5[/u] [b]p13.[/b] Given that $x^3 + y^3 = 208$ and $x + y = 4$, what is the value of $\frac{1}{x} +\frac{1}{y}$? [b]p14.[/b] Find the sum of all three-digit integers $n$ such that the value of $n$ is equal to the sum of the factorials of $n$’s digits. [b]p15.[/b] Three christmas lights are initially off. The Grinch decides to fiddle around with the lights, switching one of the lights each second. He wishes to get every possible combination of lights. After how many seconds can the Grinch complete his task? [u]Round 6[/u] [b]p16.[/b] A regular tetrahedron of side length $1$ has four similar tetrahedrons of side length $1/2$ chopped off, one from each of the four vertices. What is the sum of the numbers of vertices, edges, and faces of the remaining solid? [b]p17.[/b] Mario serves a pie in the shape of a regular $2013$-gon. To make each slice, he must cut in a straight line starting from one vertex and ending at another vertex of the pie. Every vertex of a slice must be a vertex of the original $2013$-gon. If every person eats at least one slice of pie regardless of the size, what is the maximum number of people the $2013$-gon pie can feed? [b]p18.[/b] Find the largest integer $x$ such that $x^2 + 1$ divides $x^3 + x - 1000$. [u]Round 7[/u] [b]p19.[/b] In $\vartriangle ABC$, $\angle B = 87^o$, $\angle C = 29^o$, and $AC = 37$. The perpendicular bisector of $\overline{BC}$ meets $\overline{AC}$ at point $T$. What is the value of $AB + BT$? [b]p20.[/b] Consider the sequence $f(1) = 1$, $f(2) = \frac12$ ,$ f(3) =\frac{1+3}{2}$, $f(4) =\frac{ 1+3}{2+4}$ ,$ f(5) = \frac{ 1+3+5}{2+4} . . . $ What is the minimum value of $n$, with $n > 1$, such that $|f(n) - 1| \le \frac{1}{10 }$. [b]p21.[/b] Three unit circles are centered at $(0, 0)$,$(0, 2)$, and $(2, 0)$. A line is drawn passing through $(0, 1)$ such that the region inside the circles and above the line has the same area as the region inside the circles and below the line. What is the equation of this line in $y = mx + b$ form? [u]Round 8[/u] [b]p22.[/b] The two walls of a pinball machine are positioned at a $45$ degree angle to each other. A pinball, represented by a point, is fired at a wall (but not at the intersection of the two walls). What is the maximum number of times the ball can bounce off the walls? [b]p23.[/b] Albert is fooling people with his weighted coin at a carnival. He asks his guests to guess how many times heads will show up if he flips the coin $4$ times. Richard decides to play the game and guesses that heads will show up $2$ times. In the previous game, Zach guessed that the heads would show up 3 times. In Zach’s game, there were least 3 heads, and given this information, Zach had a $\frac49$ chance of winning. What is the probability that Richard guessescorrectly? [b]p24.[/b] Let $S$ be the set of all positive integers relatively prime to $2013$ that have no prime factor greater than $15$. Find the sum of the reciprocals of all of the elements of $S$. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].