Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.

There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$. Find $p+q$.

[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. [img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img] [b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.) [b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle? [b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$. [b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $

Find all $n>1$ and integers $a_1,a_2,\dots,a_n$ satisfying the following three conditions: (i) $2<a_1\le a_2\le \cdots\le a_n$ (ii) $a_1,a_2,\dots,a_n$ are divisors of $15^{25}+1$. (iii) $2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)$

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

[b]p1.[/b] Find the sum of all $3$-digit positive integers $\overline{abc}$ that satisfy $$\overline{abc} = {n \choose a}+{n \choose b}+ {n \choose c}$$ for some $n \le 10$. [b]p2.[/b] Feng and Trung play a game. Feng chooses an integer $p$ from $1$ to $90$, and Trung tries to guess it. In each round, Trung asks Feng two yes-or-no questions about $p$. Feng must answer one question truthfully and one question untruthfully. After $15$ rounds, Trung concludes there are n possible values for $p$. What is the least possible value of $n$, assuming Feng chooses the best strategy to prevent Trung from guessing correctly? [b]p3.[/b] A hypercube $H_n$ is an $n$-dimensional analogue of a cube. Its vertices are all the points $(x_1, .., x_n)$ that satisfy $x_i = 0$ or $1$ for all $1 \le i \le n$ and its edges are all segments that connect two adjacent vertices. (Two vertices are adjacent if their coordinates differ at exactly one $x_i$ . For example, $(0,0,0,0)$ and $(0,0,0,1)$ are adjacent on $H_4$.) Let $\phi (H_n)$ be the number of cubes formed by the edges and vertices of $H_n$. Find $\phi (H_4) + \phi (H_5)$. [b]p4.[/b] Denote the legs of a right triangle as $a$ and $b$, the radius of the circumscribed circle as $R$ and the radius of the inscribed circle as $r$. Find $\frac{a+b}{R+r}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?

Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\] is a perfect cube.

Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?

In the square of an integer $ a$, the tens’ digit is $7$. What is the units’ digit of $a^2$?

Find all positive integers $n$ for which $1 - 5^n + 5^{2n+1}$ is a perfect square.

[b]p9.[/b] The frozen yogurt machine outputs yogurt at a rate of $5$ froyo$^3$/second. If the bowl is described by $z = x^2+y^2$ and has height $5$ froyos, how long does it take to fill the bowl with frozen yogurt? [b]p10.[/b] Prankster Pete and Good Neighbor George visit a street of $2021$ houses (each with individual mailboxes) on alternate nights, such that Prankster Pete visits on night $1$ and Good Neighbor George visits on night $2$, and so on. On each night $n$ that Prankster Pete visits, he drops a packet of glitter in the mailbox of every $n^{th}$ house. On each night $m$ that Good Neighbor George visits, he checks the mailbox of every $m^{th}$ house, and if there is a packet of glitter there, he takes it home and uses it to complete his art project. After the $2021^{th}$ night, Prankster Pete becomes enraged that none of the houses have yet checked their mail. He then picks three mailboxes at random and takes out a single packet of glitter to dump on George’s head, but notices that all of the mailboxes he visited had an odd number of glitter packets before he took one. In how many ways could he have picked these three glitter packets? Assume that each of these three was from a different house, and that he can only visit houses in increasing numerical order. [b]p11. [/b]The taxi-cab length of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $|x_1 - x_2| + |y_1- y_2|$. Given a series of straight line segments connected head-to-tail, the taxi-cab length of this path is the sum of the taxi-cab lengths of its line segments. A goat is on a rope of taxi-cab length $\frac72$ tied to the origin, and it can’t enter the house, which is the three unit squares enclosed by $(-2, 0)$,$(0, 0)$,$(0, -2)$,$(-1, -2)$,$(-1, -1)$,$(-2, -1)$. What is the area of the region the goat can reach? (Note: the rope can’t ”curve smoothly”-it must bend into several straight line segments.) [b]p12.[/b] Parabola $P$, $y = ax^2 + c$ has $a > 0$ and $c < 0$. Circle $C$, which is centered at the origin and lies tangent to $P$ at $P$’s vertex, intersects $P$ at only the vertex. What is the maximum value of a, possibly in terms of $c$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $ N\geq 2n \minus{} 2$ integers, of absolute value not higher than $ n > 2$, and of absolute value of their sum $ S$ less than $ n \minus{} 1,$ there exist some of sum $ 0.$ Show that for $ |S| \equal{} n \minus{} 1$ this is not anymore true, and neither for $ N \equal{} 2n \minus{} 3$ (when even for $ |S| \equal{} 1$ this is not anymore true).

[b]p1.[/b] (a) Put the numbers $1$ to $6$ on the circle in such way that for any five consecutive numbers the sum of first three (clockwise) is larger than the sum of remaining two. (b) Can you arrange these numbers so it works both clockwise and counterclockwise. [b]p2.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p3.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p4.[/b] Two consecutive three‐digit positive integer numbers are written one after the other one. Show that the six‐digit number that is obtained is not divisible by $1001$. [b]p5.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The infinite sequence $a_1,a_2,a_3,\ldots$ of positive integers is defined as follows: $a_1=1$, and for each $n \ge 2$, $a_n$ is the smallest positive integer, distinct from $a_1,a_2, \ldots , a_{n-1}$ such that: $$\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}}$$ is an integer. Prove that all positive integers appear on the sequence $a_1,a_2,a_3,\ldots$

$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$

Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$ Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]

Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers? [i]Proposed by Morteza Saghafian [/i]

In how many ways can you divide the set of eight numbers $\{2,3,\cdots,9\}$ into $4$ pairs such that no pair of numbers has $\text{gcd}$ equal to $2$?

A sequence of numbers $x_{1},x_{2},x_{3},\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50}=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

Let $A, B \in \mathbb{N}$. Consider a sequence $x_1, x_2, \ldots$ such that for all $n\geq 2$, $$x_{n+1}=A \cdot \gcd(x_n, x_{n-1})+B. $$ Show that the sequence attains only finitely many distinct values.