The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram: [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 3; /* # of vertical lines, including AB */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); } [/asy] For each $n \geq 2,$ find the number of existing parallelograms.

Evaluate \[\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}\]

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? $ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $

In a class there are n students with unequal heights. $\textbf{(a)}$ Find the number of orderings of the students such that the shortest person is not at the front and the tallest person is not at the end. $\textbf{(b)}$ Define the [i]badness[/i] of an ordering as the maximum number $k$ such that there are $k$ many people with height greater than in front of a person. For example: the sequence $66, 61, 65, 64, 62, 70$ has [i]badness [/i] $3$ since there are $3$ numbers greater than $62$ in front of it. Let $f_k(n)$ denote the number of orderings of $n$ with [i]badness[/i] $k$. Find $f_k(n)$. [hide=hint](Hint: Consider $g_k(n)$ as the number of orderings of n with [i]badness [/i]less than or equal to $k$)[/hide]

Let $P_1(x), P_2(x), \ldots, P_k(x)$ be monic polynomials of degree $13$ with integer coefficients. Suppose there are pairwise distinct positive integers $n_1, n_2, \ldots, n_k$ for which, for all positive integers $i$ and $j$ less than or equal to $k,$ the statement "$n_i$ divides $P_j(m)$ for every integer $m$" holds if and only if $i=j.$ Compute the largest possible value of $k.$

Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R^+} $ a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$ b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$

Let $S(r)$ denote the sum of the infinite geometric series $17 + 17r + 17r^2 +17r^3 + . . . $for $-1 < r < 1$. If $S(a) \times S(-a) = 2023$, find $S(a) + S(-a)$.

Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is $ \textbf{(A)}\ \frac{1}{f(x)}\qquad\textbf{(B)}\ -f(x)\qquad\textbf{(C)}\ \frac{1}{f(-x)}\qquad\textbf{(D)}\ -f(-x)\qquad\textbf{(E)}\ f(x) $

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic. Proposed by [i]Theoklitos Parayiou, Cyprus[/i]

Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact $$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?

We are given a convex quadrilateral $ABCD$ with $AD=CD$ and $\angle BAD=\angle ABC.$ Points $K$ and $L$ are middle points of $AB$ and $BC$, respectively. The rays $\overrightarrow{DL}$ and $\overrightarrow{AB}$ intersect in $M$ and the rays $\overrightarrow{DK}$ and $\overrightarrow{BC}$ – in $N$. On segment $AN$ a point $X$ is chosen, such that $AX=CM$, and on segment $AC$ – point $Y$, such that $AY=MN$. Prove that the line $AB$ bisects segment $XY$.

There is a reunion of $201$ people from $5$ different countries. It is known that in each group of $6$ people, at least two have the same age. Show that there must be at least $5$ people with the same country, age and sex.

Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area $3$, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.

The function f has the following properties : $f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$ $f(4) = 10$ Calculate $f(2001)$.

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$.

Let $\triangle ABC$ be an acute-angled triangle with altitudes $AD$ and $BE$ meeting at $H$. Let $M$ be the midpoint of segment $AB$, and suppose that the circumcircles of $\triangle DEM$ and $\triangle ABH$ meet at points $P$ and $Q$ with $P$ on the same side of $CH$ as $A$. Prove that the lines $ED, PH,$ and $MQ$ all pass through a single point on the circumcircle of $\triangle ABC$.

The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter. Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.

How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x))+f(x)+x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}?$ [i]Proposed by Kyle Lee[/i]

Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?

Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$. by Morteza Saghafian

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

Prove that the product of the sides of a quadrilateral inscribed in a circle with radius $1$ does not exceed $4$.

a) Pupils of the $8$-th form are standing in a row. There is the pupil of the $7$-th form in before each, and he is smaller (in height) than the elder. Prove that if you will sort the pupils in each of rows with respect to their height, every 8-former will still be taller than the $7$-former standing before him. b) An infantry detachment soldiers stand in the rectangle, being arranged in columns with respect to their height. Prove that if you rearrange them with respect to their height in every separate row, they will still be staying with respect to their height in columns.