Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

All diagonals of a convex polygon are drawn. Prove that its sides and diagonals can be assigned arrows in such a way that no round trip along sides and diagonals is possible.

In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called [b]uniform[/b] if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?

During a fight, each of the $2001$ roosters has torn out exactly one feather of another rooster, and each rooster has lost a feather. It turned out that among any three roosters there is one who hasn’t torn out a feather from any of the other two roosters. Find the smallest $k$ with the following property: It is always possible to kill $k$ roosters and place the rest into two henhouses in such a way that no two roosters, one of which has torn out a feather from the other one, stay in the same henhouse.

The alphabet of language $BAU$ consists of letters $B, A$, and $U$. Independently of the choice of the $BAU$ word of length n from which to start, one can construct all the $BAU$ words with length n using iteratively the following rules: (1) invert the order of the letters in the word; (2) replace two consecutive letters: $BA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU$ or $AA \to UB$. Given that $BBAUABAUUABAUUUABAUUUUABB$ is a $BAU$ word, does $BAU$ have a) the word $BUABUABUABUABAUBAUBAUBAUB$ ? b) the word $ABUABUABUABUAUBAUBAUBAUBA$ ?

The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then the number of possible coverings is even. [i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle?

Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

On the Alphamegacentavra planet there are $2023$ cities, some of which are connected by non-directed flights. It turned out that among any $4$ cities one can find two with no flight between them. Find the maximum number of triples of cities such that between any two of them there is a flight.

$T$ is a given triangle with vertices $P_1,P_2,P_3$. Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules: $(\text{i})$ If $V$ = $P_i$, then $n(V) = i$. $(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$, then $n(V) = i$ or $j$. $(\text{iii})$ If $V$ lies inside the triangle $T$, then $n(V)$ is any of the numbers $1,2,3$. Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$.

Let $\oplus$ denote the xor binary operation. Define $x \star y=(x+y)-(x\oplus y).$ Compute $$\sum_{k=1}^{63} (k \star 45).$$([i]Remark:[/i] The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2=1001_2=9.$)

In Tigre there are $2024$ islands, some of them connected by a two-way bridge. It is known that it is possible to go from any island to any other island using only the bridges (possibly several of them). In $k$ of the islands there is a flag ($0 \le k \le 2024$). Ana wants to destroy some of the bridges in such a way that after doing so, the following two conditions are met: \\ $\bullet$ If an island has a flag, it is connected to an odd number of islands. \\ $\bullet$ If an island does not have a flag, it is connected to an even number of islands. \\ Determine all values of $k$ for which Ana can always achieve her objective, no matter what the initial bridge configuration is and which islands have a flag.

A robot starts at the origin of the Cartesian plane. At each of $10$ steps, he decides to move $ 1$ unit in any of the following directions: left, right, up, or down, each with equal probability. After $10$ steps, the probability that the robot is at the origin is $\frac{n}{4^{10}}$ . Find$ n$

We will call a [i]hedgehog[/i] a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$). For each edge $e$, we denote by $s(e)$ the size of the maximum hedgehog in graph $G$, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$): \[\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.\] [i]Proposed by D. Malec, C. Tompkins[/i]

Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$. (a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times. (b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.

There's infinity of the following blocks on the table:$1*1 , 1*2 , 1*3 ,.., 1*n$. We have a $n*n$ table and Ali chooses some of these blocks so that the sum of their area is at least $n^2$. Then , Amir tries to cover the $n*n$ table so that none of blocks go out of the table and they don't overlap and he wanna maximize the covered area in the $n*n$ table with those blocks chosen by Ali. Let $k$ be the maximum coverable area independent of Ali's choice. Prove that: $$n^2 - \lceil \frac{n^2}{4} \rceil \leq k \leq n^2 - \lfloor \frac{n^2}{8} \rfloor$$ *Note : the blocks can be placed only vertically or horizontally.

Kelvin the frog is on the bottom-left lily pad of a $3 \times 3$ grid of lily pads, and his home is at the top right lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove $4$ of the lily pads so that the bottom-left and top-right lily pads both remain, but Kelvin cannot get home.

Side of an equilatreal triangle has the length $n\in\mathbb{N}.$ Each side is divided in $ n $ equal segments by division points. A line parallel with the third side of the triangle is drawn through the division points of every two sides. Let $c_n$ be the number of all rhombuses with sidelength $1$ inside the initial triangle. Prove that the greatest solution $ n $ of the inequation $c_n<2009$ is a prime number.

rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?

Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Matt has a twenty dollar bill and buys two items worth $\$7:99$ each. How much change does he receive, in dollars? [b]p2.[/b] The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers? [b]p3.[/b] Evaluate $$\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}.$$ [b]p4.[/b] A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$. [b]p5.[/b] Yannick ran $100$ meters in $14.22$ seconds. Compute his average speed in meters per second, rounded to the nearest integer. [b]p6.[/b] The mean of the numbers $2, 0, 1, 5,$ and $x$ is an integer. Find the smallest possible positive integer value for $x$. [b]p7.[/b] Let $f(x) =\sqrt{2^2 - x^2}$. Find the value of $f(f(f(f(f(-1)))))$. [b]p8.[/b] Find the smallest positive integer $n$ such that $20$ divides $15n$ and $15$ divides $20n$. [b]p9.[/b] A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ . [b]p10.[/b] Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl? [b]p11.[/b] Let $k$ be a positive integer. After making $k$ consecutive shots successfully, Andy's overall shooting accuracy increased from $65\%$ to $70\%$. Determine the minimum possible value of $k$. [b]p12.[/b] In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $ \angle AMN = \angle MNP = 90^o$. Compute the ratio $\frac{AP}{PB}$ . [b]p13.[/b] Meena writes the numbers $1, 2, 3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written? [b]p14.[/b] Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube. [b]p15.[/b] A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons? [b]p16.[/b] All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}-1$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. [b]p17.[/b] Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $y - x$. [b]p18.[/b] In triangle $ABC$, where $AC > AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are $5$ and $6$, respectively, compute the area of triangle $ABC$. [b]p19.[/b] For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x, y \le 10$ is $(x + y)^2 + (xy - 1)^2$ a prime number? [b]p20.[/b] A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.

Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length $10$, where each letter is either $A$ or $B$. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word $ABB$ is located consecutively, then these letters can be changed to $BBA$ or if $BBA$ is located somewhere in the word consecutively, then these letters can be changed to $ABB$ (if wanted, the word can be kept as it is, without making this change.) For example $AABBA$ can be transformed into $AAABB$ (the opposite is also possible.) Now Sabbir made a team of $N$ citizens where no one is friends with anyone. What is the highest value of $N.$

Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions: 1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square. 2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square. Find the maximum value of the number of the “good square”.