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

A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.

For what integers $n > 1$ can it happen that in a group of $n +1$ girls and $n$ boys, all the girls know a different number of boys while all the boys know the same number of girls? (NB Vassiliev)

Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.

Each cell in a \( 2024 \times 2024 \) table contains the letter \( A \) or \( B \), with the number of \( A \)'s in each row being the same and the number of \( B \)'s in each column being the same. Alexandra and Boris play the following game, alternating turns, with Alexandra going first. On each turn, the player chooses a row or column and erases all the letters in it that have not yet been erased, as long as at least one letter is erased during the turn, and at the end of the turn, at least one letter remains in the table. The game ends when exactly one letter remains in the table. Alexandra wins the game if the letter is \( A \), and Boris wins if it is \( B \). What is the number of initial tables for which Alexandra has a winning strategy?

A number of flowers are distributed between $n$ persons so that the first of them, Andreas, gets one flower, the other gets two flowers, the third gets three flowers, etc., to $n$-th person who gets $n$ flowers. Andreas then walks around shaking hands with each other of the others, in any order. In order to do so, he receives a flower from everyone which he hangs on to and which has more flowers than himself at the moment they shake hands. Which is the smallest number of flowers Andreas can have after shaking hands with everyone?

A regular hexagon is split into $6$ congruent equilateral triangles by drawing in the $3$ main diagonals. Each triangle is colored $1$ of $4$ distinct colors. Rotations and reflections of the figure are considered nondistinct. Find the number of possible distinct colorings.

An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n \equal{} 5$. Determine the value of $ f(2005)$.

Ten students are seated around a circular table. The teacher has a list of fifteen problems and each student is given six problems, in such a way that each problem is given exactly four times and any two students they have at most three problems in common. Prove that no matter how the teacher distributes the problems, there will always be two students sitting next to each other who have at least one problem in common.

Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have: $$|a_1|+|a_i|\leq2$$

In a $5 \times 9$ checkered rectangle, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Given is natural number $n$. Sasha claims that for any $n$ rays in space, no two of which have a common point, he will be able to mark on these rays $k$ points lying on one sphere. What is the largest $k$ for which his statement is true?

Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n$ be distinct positive real numbers. For any $(i,j)$ in a country consisting of cities $C_1,C_2,\cdots,C_n$, there is a two-way flight between $C_i$ and $C_j$ that costs $a_i+a_j$.A traveler travels between cities of this country such that every time they pay a strictly higher cost than their previous flight. Find the maximum number of flight this traveler could take.

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?

A nonempty set $S$ is called [i]Bally[/i] if for every $m\in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of Bally subsets of $\{1, 2, . . . , 2020\}$.

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$

Compute the number of ways there are to assemble $2$ red unit cubes and $25$ white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly $4$ faces of the larger cube. (Rotations and reflections are considered distinct.)

The CMU Kiltie Band is attempting to crash a helicopter via grappling hook. The helicopter starts parallel (angle $0$ degrees) to the ground. Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x$ degrees with the ground. Causing the helicopter to tilt to $90$ degrees or beyond will crash the helicopter. Find the expected number of times the band must pull the hook in order to crash the helicopter. [i]Proposed by Justin Hsieh[/i]

On a chessboard ($8\times 8$ squares with sides of length $1$) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths $1$ and $2$?

A permutation $\sigma$ of the numbers $1,2,\ldots , 10$ is called $\textit{bad}$ if there exist integers $i, j, k$ which satisfy \[1 \leq i < j < k \leq 10 \quad \text{ and }\quad \sigma(j) < \sigma(k) < \sigma(i)\] and $\textit{good}$ otherwise. Find the number of $\textit{good}$ permutations.

Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors. A move consists of an operation of one of the following three forms: [list] [*] If a duck picking rock sits behind a duck picking scissors, they switch places. [*] If a duck picking paper sits behind a duck picking rock, they switch places. [*] If a duck picking scissors sits behind a duck picking paper, they switch places. [/list] Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take place, over all possible initial configurations.

What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. To each of the thirteen points marked are assigned a color: green or red. Prove that there will always be three points of the same color that are vertices of an equilateral triangle. [img]https://cdn.artofproblemsolving.com/attachments/b/f/c50a1f8cb81ea861f16a6a47c3b758c5993213.png[/img]

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.