The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.

Could the three-dimensional space be expressed as the union of disjoint circumferences?

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

The width of a lane in a circular running track is $1.22$ meters. One loop in the first lane (shortest lane) is $400$ meters. Thus $12.5$ loops makes it a $5{,}000$ meter distance. Which lane should an athlete run in if they want to make $12$ loops as close to the $5{,}000$ meter distance as possible? \[\rm a. ~second\qquad \mathrm b. ~third \qquad \mathrm c. ~fourth \qquad\mathrm d. ~fifth \qquad\mathrm e. ~sixth\]

Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$.

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

In a class of seven students, a poll is conducted. The poll asks $~$(a) Were you born after $2021$? $~$(b) Were you born after $2019$? $~$(c) Were you born in $2020$? Five students respond "yes" to question (a), and four students respond "no" to question (b). If everyone truthfully answers all questions, how many students responded "yes" to (c)? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Given are two disjoint sets $ A$ and $ B$ such that their union is $ \mathbb N$. Prove that for all positive integers $ n$ there exist different numbers $ a$ and $ b$, both greater than $ n$, such that either $ \{ a,b,a \plus{} b \}$ is contained in $ A$ or $ \{ a,b,a \plus{} b \}$ is contained in $ B$.

Let $x$, $y$, and $z$ be real numbers satisfying the system of equations \begin{align*} xy+4z&=60\\ yz+4x&=60\\ zx+4y&=60. \end{align*} Let $S$ be the set of possible values of $x$. Find the sum of the squares of the elements of $S$.

There are $n$ passengers in a line, waiting to board a plane with $n$ seats. For $1 \le k \le n$, the $k^{th}$ passenger in line has a ticket for the $k^{th}$ seat. However, the rst passenger ignores his ticket, and decides to sit in a seat at random. Thereafter, each passenger sits as follows: If his/her assigned is empty, then he/she sits in it. Otherwise, he/she sits in an empty seat at random. How many different ways can all $n$ passengers be seated?

The numbers from 1 to $ 2013^2 $ are written row by row into a table consisting of $ 2013 \times 2013 $ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $ 1, 4, 9, \cdots, 2013^2 $ are simultaneously deleted. How many cells remain?

Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that: $x^p + y^p = p^z$

In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.

The finite sequence $a_1, a_2, ... , a_n$ of ones and zeroes should satisfy a condition: [i]for every $k$ from $0$ to $(n-1)$ the sum a_1a_{k+1} + a_2a_{k+2} + ... + a_{n-k}a_n should be odd.[/i] a) Construct such a sequence for $n=25$. b) Prove that there exists such a sequence for some $n > 1000$.

For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$ entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$.

Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.

Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$, and $C_{1}$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$, and $B_{1}$ and $Q$ lie on opposite sides of line $AC$. Prove that the points $B_{1}$, $C_{1}$, $P$, and $Q$ lie on a circle.

Let $M$ be a point on side $BC$ of isosceles $ABC$ ($AB=AC$) and let $N$ be a points on the extension of $BC$ such that $(AM)^2+(AN)^2=2(AB)^2$. Find the locus of point $N$ when point $M$ moves on side $BC$.

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $$ \int_{0}^{x} P(t)\sin t \, dt =0, \;\;\;\; \int_{0}^{x} P(t) \cos t \, dt =0 $$ has only finitely many solutions $x.$

Let $\{a_n\}$ be a sequence of real numbers with $a_1=-1$, $a_2=2$ and for all $n\ge3$, $$a_{n+1}-a_n-a_{n+2}=0.$$ Find $a_1+a_2+a_3+\ldots+a_{2015}$.

At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions left (for example, the list 2, 3, 4, 6 closes right after we add 12). Which is the maximum numbers which can be written on a list that had closed, if the list had at the beginning 10 numbers?

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/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]p4.[/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? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a board consisting of $n\times n$ unit squares where $n \ge 2$. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain $k$ tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists. (a) Find the minimal $k$ such that the game does not end for any starting configuration and choice of cells during the game. (b) Find the maximal $k$ such that the game ends for any starting configuration and choice of cells during the game. Proposed by Theresia Eisenkölbl

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]