In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N+1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1=y_2,$ $x_2=y_1,$ $x_3=y_4,$ $x_4=y_5,$ and $x_5=y_3.$ Find the smallest possible value of $N.$

Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).

We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin? [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits. Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits. How many foxes, wolves and bears were there in the hunting company?

Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.

Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.

The George Washington Bridge is $2016$ meters long. Sally is standing on the George Washington Bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\dfrac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?

In an acute-angled triangle $ABC$, point $I$ is the center of the inscribed circle, point $T$ is the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$. It turned out that $\angle AIT = 90^o$ . Prove that $AB + AC = 3BC$. (Matthew of Kursk)

A weird calculator has a numerical display and only two buttons, $\boxed{D\sharp}$ and $\boxed{D\flat}$. The first button doubles the displayed number and then adds $1$. The second button doubles the displayed number and then subtracts $1$. For example, if the display is showing $5$, then pressing the $\boxed{D\sharp}$ produces $11$. If the display shows $5$ and we press $\boxed{D\flat}$, we get $9$. If the display shows $5$ and we press the sequence $\boxed{D\sharp}$, $\boxed{D\flat}$, $\boxed{D\sharp}$, $\boxed{D\sharp}$, we get a display of $87$. [list=i] [*] Suppose the initial displayed number is $1$. Give a sequence of exactly eight button presses that will result in a display of $313$. [*] Suppose the initial displayed number is $1$, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced. [/list]

Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$. (a) Show that $P$ is the circumcenter of $BDE$. (b) Show that $A, C, D, E$ are concyclic.

Let $ a$, $ b$, $ c$, and $ d$ be real numbers with $ |a\minus{}b|\equal{}2$, $ |b\minus{}c|\equal{}3$, and $ |c\minus{}d|\equal{}4$. What is the sum of all possible values of $ |a\minus{}d|$? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

In triangle $ABC$, points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$. Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that $PB = 5$, compute the length of $BC$.

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

On the map, the Flower City has the form of a right triangle $ABC$ (see Fig.1). The length of each leg is $6$ meters. All the streets of the city run parallel to one of the legs at a distance of $1$ meter from each other. A river flows along the hypotenuse. From their houses that are located at points $V$ and $S$, at the same time get the Cog and Tab. Each short moves to rivers according to the following rule: tosses his coin, and if the [b]heads[/b] falls, he passes $1$ meter parallel to the leg $AB$ to the north (up), and if tails, then passes $1$ meter parallel to the leg $AC$ on east (right). If the Cog and the Tab meet at the same point, then they move together, tossing a coin. a) Which is more likely: Cog and Tab will meet on the way to the river, or will they come to different points on the shore? b) At what point near the river should the Stranger sit, if he wants the most did Gvintik and Shpuntik come to him together? [img]https://cdn.artofproblemsolving.com/attachments/d/c/5d6f75d039e8f2dd6a0ddfe6c4cb046b83f24c.png[/img] [hide=original wording] На мапi Квiткове мiсто має вигляд прямокутного трикутника ABC (див. рисунок 1). Довжина кожного катету – 6 метрiв. Всi вулицi мiста проходять паралельно одному за катетiв на вiдстанi 1 метра одна вiд одної. Вздовж гiпотенузи тече рiка. Зi своїх будиночкiв, що знаходяться в точках V та S, одночасно виходять Гвинтик та Шпунтик. Кожен коротулька рухається до рiчки за таким правилом: пiдкидає свою монетку, та якщо випадає Орел, вiн проходить 1 метр паралельно катету AB на пiвнiч (вгору), а якщо Решка, то проходить 1 метр паралельно катету AC на схiд (вправо). Якщо Гвинтик та Шпунтик зустрiчаються в однiй точцi, то далi вони рушають разом, пiдкидаючи монетку Гвинтика. 1. Що бiльш ймовiрно: Гвинтик та Шпунтик зустрiнуться на шляху до рiки, або вони прийдуть у рiзнi точки берега? 2. В якiй точцi бiля рiки має сидiти Незнайка, якщо вiн хоче, щоб найбiльш ймовiрно до нього прийшли Гвинтик та Шпунтик разом?[/hide]

Let $a,b$ be positive even integers. A rectangle with side lengths $a$ and $b$ is split into $a \cdot b$ unit squares. Anja and Bernd take turns and in each turn they color a square that is made of those unit squares. The person that can't color anymore, loses. Anja starts. Find all pairs $(a,b)$, such that she can win for sure. [b]Extension:[/b] Solve the problem for positive integers $a,b$ that don't necessarily have to be even. [b]Note:[/b] The [i]extension[/i] actually was proposed at first. But since this is a homework competition that goes over three months and some cases were weird, the problem was changed to even integers.

A positive integer is called [i]oneic[/i] if it consists of only $1$'s. For example, the smallest three oneic numbers are $1$, $11$, and $111$. Find the number of $1$'s in the smallest oneic number that is divisible by $63$.