Is there a $1987$-gon with consecutive sides lengths $1, 2, 3,..., 1986, 1987$, in which you can fit a circle?

Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call an integer $k \geq 1$ having property $P$, if there exists at least one integer $m \geq 1$ which cannot be expressed in the form $m = \varepsilon_1 z_1^k + \varepsilon_2 z_2^k + \cdots + \varepsilon_{2k} z_{2k}^k $ , where $z_i$ are nonnegative integer and $\varepsilon _i = 1$ or $-1$, $i = 1, 2, \ldots, 2k$. Prove that there are infinitely many integers $k$ having the property $P.$

We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$. If the chosen sequence is $a_1, a_2, ..., a_k$ ($k$ is not fixed), then at the $i^{th}$ step, we choose the first a$_i$ numbers on the line and change the 1s to 0s and 0s to 1s. After $k$ steps are over, we calculate the sum of the numbers on the board, say $S$. The expected value of $S$ is $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a + b.$

Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.

As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour. Ermenson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Ermenson? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 $

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\] is a perfect cube.

Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls?

10.4, 11.3 Given $N\geq 3$ points enumerated with 1, 2, ..., $N$. Each two numbers are connected by mean of arrow from a lesser number to a greater one. A coloring of all arrows into red and blue is called [i]monochromatic[/i] iff for any numbers $A$ and $B$ there are [color=red]no[/color] two monochromatic paths from $A$ to $B$ of different colors. Find the number of monochromatic colorings. ([i]I. Bogdanov, G. Chelnokov[/i])

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Given the natural $n$. We shall call [i]word [/i] sequence from $n$ letters of the alphabet, and [i]distance [/i] $\rho(A, B)$ between [i]words [/i] $A=a_1a_2\dots a_n$ and $B=b_1b_2\dots b_n$ , the number of digits in which they differ (that is, the number of such $i$, for which $a_i\ne b_i$). We will say that the [i]word [/i] $C$ [i]lies [/i] between words $A$ and $B$ , if $\rho (A,B)=\rho(A,C)+\rho(C,B)$. What is the largest number of [i]words [/i] you can choose so that among any three, there is a [i]word lying[/i] between the other two?

A circle is divided by $2n$ points into $2n$ equal arcs. Let $P_1, P_2, \ldots, P_{2n}$ be an arbitrary permutation of the $2n$ division points. Prove that the polygonal line $P_1 P_2 \cdots P_{2n} P_1$ contains at least two parallel segments.

In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color. [b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b] In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color. ([i]Edited by Orlando Döhring[/i]) [i]Author: Radu Gologan and Dan Schwarz, Romania[/i]

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that: $$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$

You have $128$ teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the $128$ teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the $128$ teams are randomly put into the bracket. What is the probability that the Engineers play the Crimson sometime during the tournament?

For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$ Find the value of $\frac{I_1}{I_2}.$

A jar contains $97$ marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\tfrac{5}{12}.$ After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.

In the acute triangle $ABC$ the side $BC> AB$, and the angle bisector $BL = AB$. On the segment $BL$ there is a point $M$, for which $\angle AML = \angle BCA$. Prove that $AM = LC$.

The diagonals of the inscribed quadrilateral $ABCD$ meet at the point $M$, $\angle AMB = 60^o$. Equilateral triangles $ADK$ and $BCL$ are built outward on sides $AD$ and $BC$. Line $KL$ meets the circle circumscribed ariound $ABCD$ at points $P$ and $Q$. Prove that $PK = LQ$.

What is the hybridization of the carbon atom in a carboxyl group? $ \textbf{(A) }sp \qquad\textbf{(B) } sp^2\qquad\textbf{(C) } sp^3\qquad\textbf{(D) } dsp^3\qquad $