Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition $$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$ Thanks @dgrozev

Is it possible to fill a $40 \times 41$ table with integers so that each integer equals the number of adjacent (by an edge) cells with the same integer? Alexandr Gribalko

Prove that for any set containing $2047$ positive integers, there exists $1024$ positive integers in the set such that the sum of these positive integers is divisible by $1024$.

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

A square is divided into $16$ equal squares, obtaining the set of $25$ different vertices. What is the least number of vertices one must remove from this set, so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?

A cube stands on one of the squares of an infinite chessboard. On each face of the cube there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks on the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. Prove that the cube cannot cover any $5\times 5$ square.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

[u]Set 5[/u] [b]p13.[/b] An equiangular $12$-gon has side lengths that alternate between $2$ and $\sqrt3$. Find the area of the circumscribed circle of this $12$-gon. [b]p14.[/b] For positive integers $n$, let $\sigma(n)$ denote the number of positive integer factors of $n$. Then $\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64$. Let $S$ be the set of factors $k$ of $17280$ such that $\sigma(k) = 32$. If $p$ is the product of the elements of $S$, find $\sigma(p)$. [b]p15.[/b] How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid. PS. You should use hide for answers.

A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points

[u]Set 4[/u] [b]p16.[/b] Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered $1-50$ each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest value or is tied for the highest value of all $4$ drawn cards? [b]p17.[/b] Suppose that $s$ is the sum of all positive values of $x$ that satisfy $2016\{x\} = x+[x]$. Find $\{s\}$. (Note: $[x]$ denotes the greatest integer less than or equal to $x$ and $\{x\}$ denotes $x - [x]$.) [b]p18.[/b] Let $ABC$ be a triangle such that $AB = 41$, $BC = 52$, and $CA = 15$. Let H be the intersection of the $B$ altitude and $C$ altitude. Furthermore let $P$ be a point on $AH$. Both $P$ and $H$ are reflected over $BC$ to form $P'$ and $H'$ . If the area of triangle $P'H'C$ is $60$, compute $PH$. [b]p19.[/b] A random integer $n$ is chosen between $1$ and $30$, inclusive. Then, a random positive divisor of $n, k$, is chosen. What is the probability that $k^2 > n$? [b]p20.[/b] What are the last two digits of the value $3^{361}$? [u]Set 5[/u] [b]p21.[/b] Let $f(n)$ denote the number of ways a $3 \times n$ board can be completely tiled with $1 \times 3$ and $1 \times 4$ tiles, without overlap or any tiles hanging over the edge. The tiles may be rotated. Find $\sum^9_{i=0} f(i) = f(0) + f(1) + ... + f(8) + f(9)$. By convention, $f(0) = 1$. [b]p22.[/b] Find the sum of all $5$-digit perfect squares whose digits are all distinct and come from the set $\{0, 2, 3, 5, 7, 8\}$. [b]p23.[/b] Mary is flipping a fair coin. On average, how many flips would it take for Mary to get $4$ heads and $2$ tails? [b]p24.[/b] A cylinder is formed by taking the unit circle on the $xy$-plane and extruding it to positive infinity. A plane with equation $z = 1 - x$ truncates the cylinder. As a result, there are three surfaces: a surface along the lateral side of the cylinder, an ellipse formed by the intersection of the plane and the cylinder, and the unit circle. What is the total surface area of the ellipse formed and the lateral surface? (The area of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is $\pi ab$.) [b]p25.[/b] Let the Blair numbers be defined as follows: $B_0 = 5$, $B_1 = 1$, and $B_n = B_{n-1} + B_{n-2}$ for all $n \ge 2$. Evaluate $$\sum_{i=0}^{\infty} \frac{B_i}{51^i}= B_0 +\frac{B_1}{51} +\frac{B_2}{51^2} +\frac{B_3}{51^3} +...$$ [u]Estimation[/u] [b]p26.[/b] Choose an integer between $1$ and $10$, inclusive. Your score will be the number you choose divided by the number of teams that chose your number. [b]p27.[/b] $2016$ blind people each bring a hat to a party and leave their hat in a pile at the front door. As each partier leaves, they take a random hat from the ones remaining in a pile. Estimate the probability that at least $1$ person gets their own hat back. [b]p28.[/b] Estimate how many lattice points lie within the graph of $|x^3| + |y^3| < 2016$. [b]p29.[/b] Consider all ordered pairs of integers $(x, y)$ with $1 \le x, y \le 2016$. Estimate how many such ordered pairs are relatively prime. [b]p30.[/b] Estimate how many times the letter “e” appears among all Guts Round questions. PS. You should use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c3h2779594p24402189]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the following figure, it is desired to go from point \( A \) to point \( B \) by walking only along the lines of the figure up and to the right. How many different paths can we take? [img]https://cdn.artofproblemsolving.com/attachments/7/c/dce2e0bcb69c9e2014474ab3699b4ef0470497.png[/img]

For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?

There is a board with $n$ rows and $4$ columns, and white, yellow and light blue chips. Player $A$ places four tokens on the first row of the board and covers them so Player $B$ doesn't know them. How should player $B$ do to fill the minimum number of rows with chips that will ensure that in any of the rows he will have at least three hits? Clarification: A hit by player $B$ occurs when he places a token of the same color and in the same column as $A$.

Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to finish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?

Albert and Barbara play the following game. On a table there are $1999$ sticks, and each player in turn removes some of them: at least one stick, but at most half of the currently remaining sticks. The player who leaves just one stick on the table loses the game. Barbara moves first. Decide which player has a winning strategy and describe that strategy.

Nine people have attended four different meetings sitting around a circular table. Could they have done so in a way that no two people sat next to each other more than once? Justify your answer.

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$. a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$. b) does it always happen that there exists a right triangle with the above property ?

Let $n\geq 2$ be an integer. Let us call [i]interval[/i] a subset $A \subseteq \{1,2,\ldots,n\}$ for which integers $1\leq a < b\leq n$ do exist, such that $A = \{a,a+1,\ldots,b-1,b\}$. Let a family $\mathcal{A}$ of subsets $A_i \subseteq \{1,2,\ldots,n\}$, with $1\leq i \leq N$, be such that for any $1\leq i < j \leq N$ we have $A_i \cap A_j$ being an interval. Prove that $\displaystyle N \leq \left \lfloor n^2/4 \right \rfloor$, and that this bound is sharp. (Dan Schwarz - after an idea by Ron Graham)

We are given $m\times n$ table tiled with $3\times 1$ stripes and we are given that $6 | mn$. Prove that there exists a tiling of the table with $2\times 1$ dominoes such that each of these stripes contains one whole domino.

38th Austrian Mathematical Olympiad 2007, round 3 problem 5 Given is a convex $ n$-gon with a triangulation, that is a partition into triangles through diagonals that don’t cut each other. Show that it’s always possible to mark the $ n$ corners with the digits of the number $ 2007$ such that every quadrilateral consisting of $ 2$ neighbored (along an edge) triangles has got $ 9$ as the sum of the numbers on its $ 4$ corners.

Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a [i]clique[/i] if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its [i]size[/i]. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room. [i]Author: Vasily Astakhov, Russia[/i]

Show that the number of finite sequences of length $ n $ that are formed with $ 5 $ distinct numbers such that for any three consecutive terms of the sequence at least two are equal, is $ \frac{5}{2}\left( (-1)^n+3^n \right) . $

The integers $1$ through $1000$ are located on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e.,$1, 16, 31, \cdots$ ) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?