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 $N.$

Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased. [i]Proposed by Ankan Bhattacharya[/i]

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

Let $A_{1}A_{2}\ldots A_{9}$ be a regular $9-$gon. Let $\{A_{1},A_{2},\ldots,A_{9}\}=S_{1}\cup S_{2}\cup S_{3}$ such that $|S_{1}|=|S_{2}|=|S_{3}|=3$. Prove that there exists $A,B\in S_{1}$, $C,D\in S_{2}$, $E,F\in S_{3}$ such that $AB=CD=EF$ and $A \neq B$, $C\neq D$, $E\neq F$.

Let $a,b,c,d$ be four non-negative numbers satisfying \[ a+b+c+d=1. \] Prove the inequality \[ a \cdot b \cdot c + b \cdot c \cdot d + c \cdot d \cdot a + d \cdot a \cdot b \leq \frac{1}{27} + \frac{176}{27} \cdot a \cdot b \cdot c \cdot d. \]

Let $D$ be a regular dodecahedron, which is a polyhedron with $20$ vertices, $30$ edges, and $12$ regular pentagon faces. A tetrahedron is a polyhedron with $4$ vertices, $6$ edges, and $4$ triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of $D$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png[/img]

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies \[ 4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)} \]

Answer the following questions : $\textbf{(a)}~$ A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2,\cdots, a_k$, each $a_i>1$, such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1$$ Show that if $k$ is stable, then $(k+1)$ is also stable. Using this or otherwise, find all stable numbers. $\textbf{(b)}$ Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^*(y):=\max_{x\in A} \left\{yx-f(x)\right\}$$ whenever the above maximum is finite. For the function $f(x)=\ln x$, determine the set of points for which $f^*$ is defined and find an expression for $f^*(y)$ involving only $y$ and constants.

Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \] Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.

In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer $n$ for which any tennis tournament with $n$ people has a group of four tennis players that is ordered. [i]Ray Li[/i]

Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions: $f(f(x,y),z)=f(x,f(y,z))$; $f(x,y) = f(y,x)$; $f(x,1)=x$; $f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$

Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$

The cross-section of a right cylinder is an ellipse, with semi-axes $a$ and $b,$ where $a > b.$ The cylinder is very long, made of very light homogeneous material. The cylinder rests on the horizontal ground which it touches along the straight line joining the lower endpoints of the minor axes of its several cross-sections. Along the upper endpoints of these minor axes lies a very heavy homogeneous wire, straight and just as long as the cylinder. The wire and the cylinder are rigidly connected. We neglect the weight of the cylinder, the breadth of the wire, and the friction of the ground. The system described is in equilibrium, because of its symmetry. This equilibrium seems to be stable when the ratio $b/a$ is very small, but unstable when this ratio comes close to $1.$ Examine this assertion and find the value of the ratio $b/a$ which separates the cases of stable and unstable equilibrium.

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

In a dihedral angle of measure $c$ two non-intersecting spheres are inscribed, the centers of which are located on a straight line perpendicular to the edge of the dihedral angle. The points of contact of these spheres with the edges of the corner are at distances $a$ and $b$ from the edge. Let us consider an arbitrary plane tangent to these spheres and intersecting the segment connecting their centers. Let us denote by $\phi$ the measure of the angle formed at the intersection of this plane with the faces of a given dihedral angle. Find the greatest value $\phi$.

Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2n,2n)$ consists of $4n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0),(2n,0),(2n,2n),(0,2n)$ into 2 regions with even areas is $$\frac{{4n \choose 2n} + {2n \choose n}}{2}$$

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Determine the number of subset $S$ of the set $T = {1, 2,..., 2005}$ such that the sum of elements in $s$ is congruent to 2006 modulo 2048.

Let $z_1$, $z_2$, and $z_3$ be the complex roots of the equation $(2z -3\overline{z})^3 = 54i+54$. Compute the area of the triangle formed by $z_1$, $z_2$, and $z_3$ when plotted in the complex plane.

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that [list=a] [*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty; [*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$; [*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$ [/list] [i]Mathematical Gazette[/i]

Let $k$ and $n$ be positive integers, with $k \le n$. A certain class has n students, and among any $k$ of them there is always one that is friends with the other $k- 1$. Find all values of $k$ and $n$ for which there must necessarily be a student who is friends with everyone else in the class.

Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$