Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements. [b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]

Natural nonzero numder, which consists of $ m$ digits, is called hiperprime, if its any segment, which consists $ 1,2,...,m$ digits is prime (for example $ 53$ is hiperprime, because numbers $ 53,3,5$ are prime). Find all hiperprime numbers.

Is it possible to insert numbers $1, \ldots, 16$ into a table $4 \times 4$ (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers $a$ and $b$ satisfying\\ (a) $|a-b| \geq 6$\\ (b) $|a-b| \geq 7$

A weight of $11111$ grams is placed in the left pan of a balance. Weights are added one at a time, the first weighing $1$ gram, and each subsequent one weighing twice as much as the preceding one. Each weight may be added to either pan. After a while, equilibrium is achieved. Is the $16$ gram weight placed in the left pan or the right pan? ( AV Kalinin)

[u]Round 1[/u] [b]p1.[/b] Elaine creates a sequence of positive integers $\{s_n\}$. She starts with $s_1 = 2018$. For $n \ge 2$, she sets $s_n =\frac12 s_{n-1}$ if $s_{n-1}$ is even and $s_n = s_{n-1} + 1$ if $s_{n-1}$ is odd. Find the smallest positive integer $n$ such that $s_n = 1$, or submit “$0$” as your answer if no such $n$ exists. [b]p2.[/b] Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins? [b]p3.[/b] Four circles each of radius $\frac14$ are centered at the points $\left( \pm \frac14, \pm \frac14 \right)$, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle? [u]Round 2 [/u] [b]p4.[/b] If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing? [b]p5.[/b] Let $a_1 = 2018$, and for $n \ge 2$ define $a_n = 2018^{a_{n-1}}$ . What is the ones digit of $a_{2018}$? [b]p6.[/b] We can write $(x + 35)^n =\sum_{i=0}^n c_ix^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$? [u]Round 3[/u] [b]p7.[/b] How many positive integers are factors of $12!$ but not of $(7!)^2$? [b]p8.[/b] How many ordered pairs $(f(x), g(x))$ of polynomials of degree at least $1$ with integer coefficients satisfy $f(x)g(x) = 50x^6 - 3200$? [b]p9.[/b] On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between $1$ and $10$ (inclusive). What is the probability that the average of their three scores is an integer? [u]Round 4[/u] [b]p10.[/b] Find the largest positive integer N such that $$(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)$$ is divisible by $N$ for all choices of positive integers $a > b > c > d > e$. [b]p11.[/b] Let $ABCDE$ be a square pyramid with $ABCD$ a square and E the apex of the pyramid. Each side length of $ABCDE$ is $6$. Let $ABCDD'C'B'A'$ be a cube, where $AA'$, $BB'$, $CC'$, $DD'$ are edges of the cube. Andy the ant is on the surface of $EABCDD'C'B'A'$ at the center of triangle $ABE$ (call this point $G$) and wants to crawl on the surface of the cube to $D'$. What is the length the shortest path from $G$ to $D'$? Write your answer in the form $\sqrt{a + b\sqrt3}$, where $a$ and $b$ are positive integers. [b]p12.[/b] A six-digit palindrome is a positive integer between $100, 000$ and $999, 999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A cycling group that has $4n$ members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly $2n$ members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members. Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

In every cell of a square table is a number. The sum of the largest two numbers in each row is $a$ and the sum of the largest two numbers in each column is b. Prove that $a = b$.

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

Consider $n\geqslant 6$ coplanar lines, no two parallel and no three concurrent. These lines split the plane into unbounded polygonal regions and polygons with pairwise disjoint interiors. Two polygons are non-adjacent if they do not share a side. Show that there are at least $(n-2)(n-3)/12$ pairwise non-adjacent polygons with the same number of sides each.

In Occidentalia there are $20$ different companies, each looking to hire $15$ new employees. A group of $300$ applicants interview each of the companies. Each company qualifies each applicant as suitable or not suitable to work in it, in such a way that each of them finds exactly $p$ suitable applicants, with $p > 15$. and each applicant is found suitable by at least one company. What is the smallest of $p $f or which it is always possible to assign $15$ applicants to each company, given that each company is assigned only applicants that it considers appropriate, and that each of the $300$ applicants is assigned to a company?

Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning \[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \] Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$. [i]Yang Liu[/i]

There is a table with $n > 2$ cells in the first row, $n-1$ cells in the second row is a cell, $n-2$ in the third row, $\ldots$, $1$ cell in the $n$-th row. The cells are arranged as shown below. [img]https://i.ibb.co/0Z1CR0c/UMO24-8-2.png[/img] In each cell of the top row Petryk writes a number from $1$ to $n$, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers $a, b$, it contains number $|a-b|$. What is the largest number that can be written in a single cell of the bottom row? [i]Proposed by Bogdan Rublov[/i]

Kirill has $n{}$ identical footballs and two infinite rows of baskets, each numbered with consecutive natural numbers. In one row the baskets are red, in the other they are blue. Kirill puts all the balls into baskets so that the number of balls in the either row of baskets does not increase. Denote by $A{}$ the number of ways to arrange the balls so that the first blue basket contains more balls than any red one, and by $B{}$ the number of arrangements so that the number of some blue basket corresponds with the number of balls in it. Prove that $A = B$.

In each square of a $5\times 5$ board is written $1$ or $-1$. In each step, the number of each of the $25$ cells is replaced by the result of the multiplication of the numbers of all its neighboring cells. Initially we have the board of the figure. [img]https://cdn.artofproblemsolving.com/attachments/2/d/29b8e5df29526630102ac400a3a2b2f8fee4f7.gif[/img] Show how the board looks after $2004$ steps. Clarification: Two squares are [i]neighbors [/i] if they have a common side.

a) In each box of a $7\times 7$ board one of the numbers is written: $1, 2, 3, 4, 5, 6$ or $7$ of so that each number is written in seven different boxes. Is it possible that in no row and no column are consecutive numbers written? b) In each box of a $5\times 5$ board one of the numbers is written: $1, 2, 3, 4$ or $5$ of so that each one is written in five different boxes. Is it possible that in no row and in no column are consecutive numbers written?

Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

A square is divided into $25$ small squares, equal to each other, drawing lines parallel to the sides of the square. Some are drawn diagonals of small squares so that there are no two diagonals with a common point. What is the maximum number of diagonals that can be traced?

Let $ G(V,E)$ be a connected graph, and let $ d_G(x,y)$ denote the length of the shortest path joining $ x$ and $ y$ in $ G$. Let $ r_G(x)\equal{} \max \{ d_G(x,y) : \; y \in V \ \}$ for $ x \in V$, and let $ r(G)\equal{} \min \{ r_G(x) : \;x \in V\ \}$. Show that if $ r(G) \geq 2$, then $ G$ contains a path of length $ 2r(G)\minus{}2$ as an induced subgraph. [i]V. T. Sos[/i]

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

There are many of the same regular triangles. At the vertices of each of them, the numbers $1, 2, 3$ are written in random order. The triangles were superimposed on one another and found the sum of the numbers that fell into each of the three corners of the stack. Could it be that in each corner the sum is equal to: a) $25$, b) $50$?

[b]8.1[/b] $x$ and $y$ are the roots of the equation $t^2-ct-c=0$. Prove that holds the inequality $x^3 + y^3 + (xy)^3 \ge 0.$ [b]8.2.[/b] Two circles touch internally at point $A$ . Through a point $B$ of the inner circle, different from $A$, a tangent to this circle intersecting the outer circle at points C and $D$. Prove that $AB$ is a bisector of angle $CAD$. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png[/img] [b]8.3[/b] Prove that $2^{3^{100}} + 1$ is divisible by $3^{101}$. [b]8.4 / 7.5[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png [/img] [b]8.5[/b] In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends. [b]8.6[/b] Numbers $a_1, a_2, . . . , a_{100}$ are such that $$a_1 - 2a_2 + a_3 \le 0$$ $$a_2-2a_3 + a_ 4 \le 0$$ $$...$$ $$a_{98}-2a_{99 }+ a_{100} \le 0$$ and at the same time $a_1 = a_{100}\ge 0$. Prove that all these numbers are non-negative. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find: a) The minimum value of $n$. b) For that value, find the least possible number of red squares.