There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game? $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$

In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark. [asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$2$",(0,0)); draw(shift(1,0)*box); label("$?$",(1,0)); draw(shift(2,0)*box); label("$?$",(2,0)); draw(shift(3,0)*box); label("$?$",(3,0)); draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$5$",(1,0.8)); draw(shift(2,0.8)*box); label("$?$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

The maximum number of solid regular tetrahedrons can be placed against each other so that one of their edges coincides with a given line segment in space? [hide=original wording]Hoeveel massieve regelmatige viervlakken kan men maximaal tegen mekaar plaatsen zodat ´e´en van hun ribben samenvalt met een gegeven lijnstuk in de ruimte?[/hide]

In the Cartesian coordinate system, we define a [i]Bongo-line[/i] as a sequence of integer points $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ such that: - $A_iA_{i+1} = \sqrt{2}$ for every $i \in \mathbb{Z}$; - the polyline $\ldots A_{-1}A_0A_1 \ldots$ has no self-intersections. Let $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ and $\beta = (\ldots, B_{-1}, B_0, B_1, \ldots)$ be two Bongo-lines such that there exists a bijection $f : \mathbb{Z} \to \mathbb{Z}$ such that $A_iA_{i+1}$ and $B_{f(i)}B_{f(i)+1}$ halve each other. Prove that all vertices of $\alpha$ and $\beta$ lie on two lines. [i]Proposed by Pavle Martinović[/i]

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

Find all pairs $(a, b)$ of real numbers with the following property: [list]Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.[/list]

There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.

Assume $ a,b,c $ are arbitrary reals such that $ a+b+c = 0 $. Show that $$ \frac{33a^2-a}{33a^2+1}+\frac{33b^2-b}{33b^2+1}+\frac{33c^2-c}{33c^2+1} \ge 0 $$

Successive discounts of $10\%$ and $20\%$ are equivalent to a single discount of: $\textbf{(A)}\ 30\% \qquad \textbf{(B)}\ 15\% \qquad \textbf{(C)}\ 72\% \qquad \textbf{(D)}\ 28\% \qquad \textbf{(E)}\ \text{None of these}$

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

Given a positive integer $n$, suppose that $P(x,y)$ is a real polynomial such that \[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} \] What is the minimum degree of $P$? [i]Proposed by Loke Zhi Kin[/i]

Shen, Ling, and Ru each place four slips of paper with their name on it into a bucket. They then play the following game: slips are removed one at a time, and whoever has all of their slips removed first wins. Shen cheats, however, and adds an extra slip of paper into the bucket, and will win when four of his are drawn. Given that the probability that Shen wins can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$.

Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that \[(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)\] for all $x,y\in\mathbb{Q}$

Let $\omega_1$ and $\omega_2$ be circles with centers $O_1$ and $O_2$, respectively, and radii $r_1$ and $r_2$, respectively. Suppose that $O_2$ is on $\omega_1$. Let $A$ be one of the intersections of $\omega_1$ and $\omega_2$, and $B$ be one of the two intersections of line $O_1O_2$ with $\omega_2$. If $AB = O_1A$, find all possible values of $\frac{r_1}{r_2}$ .

Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.

Solve the system $$ \left\{\begin{matrix} ax=b\\bx=a \end{matrix}\right. $$ independently of the fixed elements $ a,b $ of a group of odd order. [i]Marian Andronache[/i]

We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?

A die is thrown six times. How many ways are there for the six rolls to sum to $21$?

$x_i\in\mathbb{R}(i=1,2,\cdots,n;n\geq2)$, satisfying that $\sum_{i=1}^n|x_i|=1,\sum_{i=1}^nx_i=0$. Prove that $|\sum_{i=1}^n\frac{x_i}{i}|\leq\frac{1}{2}-\frac{1}{2n}$

A rectangular array has $m$ rows and $n$ columns, where $m < n$. Some cells of the array contain stars, in such a way that there is at least one star in each column. Prove that there is at least one such star such that the row containing it has more stars than the column containing it. (A. Razborov, Moscow)

You have $4$ game pieces, and you play a game against an intelligent opponent who has $6$. The rules go as follows: you distribute your pieces among two points a and b, and your opponent simultaneously does as well (so neither player sees what the other is doing). You win the round if you have more pieces than them on either $a$ or$ b$, and you lose the round if you only draw or have fewer pieces on both. You play the optimal strategy, assuming your opponent will play with the strategy that beats your strategy most frequently. What proportion of the time will you win?

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.