The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$ [asy] unitsize(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.5,6)--(10,6)); draw((-0.5,7)--(10,7)); label("frequency",(-0.5,8)); label("0", (-1, 0)); label("1", (-1, 1)); label("2", (-1, 2)); label("3", (-1, 3)); label("4", (-1, 4)); label("5", (-1, 5)); label("6", (-1, 6)); label("7", (-1, 7)); filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black); filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black); filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black); filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black); label("3", (0.5, -0.5)); label("4", (2.5, -0.5)); label("5", (4.5, -0.5)); label("6", (6.5, -0.5)); label("7", (8.5, -0.5)); label("name length", (4.5,-1.5)); [/asy]

''Moskvich'' and ''Zaporozhets'' drove past the observer on the highway and the Niva moving towards them. It is known that when the Moskvich caught up with the observer, it was equidistant from the Zaporozhets and the Niva, and when the Niva caught up with the observer, it was equal. but removed from ''Moskvich'' and ''Zaporozhets''. Prove that ''Zaporozhets'' at the moment of passing by the observer was equidistant from the Niva and ''Moskvich''.

Let $ABC$ be a triangle, $X$ the midpoint of arc $BC$ on the circumcircle. The tangents from $X$ to the incircle meet the circumcircle again at $X_1,X_2$, and $X_1X_2$ intersects the incircle at $P,Q$. Let $M$ be the midpoint of $PQ$, and let $A_1$ be the tangency point of the $A$-mixtillinear incircle with the circumcircle. Show that $A,M,A_1$ are collinear.

In a graph with $2n-1$ vertices throwing out any vertex the remaining graph has a complete subgraph with $n$ vertices. Prove that the initial graph has a complete subgraph with $n+1$ vertices.

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane. Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$. Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$ for any $A, B \in S$. [i] Proposed by Victor Wang [/i]

Inside a circle, point $K$ is taken such that the ray drawn from $K$ through the centre $O$ of the circle and the chord perpendicular to this ray passing through $K$ divide the circle into three pieces with equal area. Let $L$ be one of the endpoints of the chord mentioned. Does the inequality $\angle KOL < 75^o$ hold?

Let $S$ be the center of a square $ABCD$ and $P$ be the midpoint of $AB$. The lines $AC$ and $PD$ meet at $M$, and the lines $BD$ and $PC$ meet at $N$. Prove that the radius of the incircle of the quadrilateral $PMSN$ equals $MP-MS$.

How many ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 10$ are there such that in the geometric sequence whose first term is $a$ and whose second term is $b$, the third term is an integer?

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

For \(u,v \in\mathbb{R}^4\), let \(<u,v>\) denote the usual dot product. Define a [i]vector field[/i] to be a map \(\omega:\mathbb{R}\to\mathbb{R}\) such that \(<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.\) Find a maximal collection of vector fields \(\left\{\omega_1,...,\omega_k\right\}\) such that the map \(\Omega\) sending \(z\) to \(\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z)\), with \(\lambda_1,\ldots,\lambda_k\in\mathbb{R}\), is nonzero on \(\mathbb{R}^4\backslash\{0\}\) unless \(\lambda_1=\cdots=\lambda_k=0\)

Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties: - \(a_0\) is a given positive integer; - For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square. For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on. (a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers. Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite. (b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).

Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

The number $5.6$ may be expressed uniquely (ignoring order) as a product $\underline{a}.\underline{b} \times \underline{c}.\underline{d}$ for digits $a,b,c,d$ all nonzero. Compute $\underline{a}.\underline{b}+\underline{c}.\underline{d}.$

Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $ X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA+XB+XC$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4} $

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

Let $f(x)$ be a function, where if $q$ is an integer, then $f(\tfrac{1}{q})=q,$ and if $m$ and $n$ are real numbers, $f(m\cdot n)$ $ =$ $ f(m)\cdot f(n).$ If $f(\sqrt{2})$ can be written as $\tfrac{\sqrt{a}}{b}$ where $a$ is not divisible by the square of any prime and $b$ is a positive integer, then what is $a+b$?

Two quadratic trinomials $ f (x) $ and $ g (x) $ differ from each other only by a permutation of coefficients. Could it be that $ f (x) \geq g (x) $ for all real $ x $?

Csenge and Eszter ate a whole basket of cherries. Csenge ate a quarter of all cherries while Eszter ate four-sevenths of all cherries and forty more. How many cherries were in the basket in total?

Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$

In the triangle $ABC$, points $D$ and $E$ lie on the side $BC$, with $CE = BD$. Also, $M$ is the midpoint of $AD$. Show that the centroid of $ABC$ lies on $ME$.

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

Divide the upper right quadrant of the plane into square cells with side length $1$. In this quadrant, $n^2$ cells are colored, show that there’re at least $n^2+n$ cells (possibly including the colored ones) that at least one of its neighbors are colored.

Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.