The coordinates of $ A,B$ and $ C$ are $ (5,5),(2,1)$ and $ (0,k)$ respectively. The value of $ k$ that makes $ \overline{AC}\plus{}\overline{BC}$ as small as possible is: $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac{1}{2} \qquad\textbf{(C)}\ 3\frac{6}{7} \qquad\textbf{(D)}\ 4\frac{5}{6} \qquad\textbf{(E)}\ 2\frac{1}{7}$

Fix circle with center $O$, diameter $AB$ and a point $C$ on it, different from $A, B$. Let a point $D$, different from $A, B$, vary on the arc $AB$ not containing $C$. Let $E$ lie on $CD$ such that $BE \perp CD$. Prove that $CE \cdot ED$ is maximal exactly when $BOED$ is cyclic.

Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly$ 20$ dierent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$. Find all possible values of $N$.(Both $1$ and $m$ are counted as divisors of the number $m$.)

Andy, Bella, and Chris are playing in a trivia contest. Andy has $21,200$ points, Bella has $23,600$ points, and Chris has $11,200$ points. They have reached the final round, which works as follows: [list] [*] they are given a hint as to what the only question of the round will be about; [*] then, each of them must bet some amount of their points---this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on); [*] then, they will be shown the question, where they will have $30$ seconds to individually submit a response (a player does not know any of the other players' answers); [*] finally, once time runs out, their responses and bets are revealed---if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided). [/list] Suppose that the contestants are currently deciding their bets based on the hint that the question will be about history. Bella knows that she will likely get the question wrong, but she also knows that Andy, who dislikes history, will definitely get it wrong. Knowing this, Bella wagers an amount that will guarantee her a win. Find the maximum number of points Bella could have ended up with.

Let n be a fixed prime number >3. A triangle is said to be admissible if the measure of each of its angles is of the form $\frac{m\cdot 180^{\circ}}{n}$ for some positive integer m. We are given one admissible triangle. Every minute we cut one of the triangles we already have into two admissible triangles so that no two of the triangles we have after cutting are similar. After some time, it turns out that no more cuttings are possible. Prove that at this moment, the triangles we have contain all possible admissible triangles (we do not distinguish between triangles which have same sets of angles, i.e. similar triangles).

A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero. [i]Proposed by Morteza Saghafian[/i]

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

Which pair of numbers does NOT have a product equal to $36$? $\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}$

Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$.

Let us denote by $<a>$ the distance from $a$ to the nearest integer. (For example, $<1,2> = 0.2$, $<\sqrt3> = 2-\sqrt3$) How many solutions does the system of equations have $$\begin{cases} <19x>=y \\ <96y>=x \end{cases} \,\,\, ?$$

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory. a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane. Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then $$g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1 \,\,\,\, (*)$$ Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property. b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$. c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$. d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$. e) Prove that for all $x , f(x) = g(x)$.

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

How many distinct triangles can be drawn using three of the dots below as vertices? [asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy] $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24 $

For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$. [list][*]If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$. [*]If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.[/list] Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps. [i]Proposed by Gerhard Woeginger, Austria[/i]

If $A$ and $B$ are vertices of a polyhedron, define the [i]distance[/i] $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$? $\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$

Show that for all real numbers $ a,b,c,d,$ the following inequality holds: \[ (a\plus{}b\plus{}c\plus{}d)^2 \leq 3 (a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} 6ab\]

Prove that for every positive $x_1, x_2, x_3, x_4, x_5$ holds inequality: $$(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_3x_4 + x_5x_1 + x_2x_3 + x_4x_5)$$

Find the sum of all three digit numbers (written in base $10$) such that the leading digit is the sum of other two digits. Express your answer in base $10$.

$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to $\textbf{(A) }2\qquad\textbf{(B) }2\sqrt{3}\qquad\textbf{(C) }4\sqrt{2}\qquad\textbf{(D) }\sqrt{6}\qquad \textbf{(E) }2\sqrt{2}$

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

There are balls that look identical, but their weights all dier by a little. We have a balance that can compare only two balls at a time. What is the minimum number of times, in the worst case, we have to use to balance to rank all balls by weight?

In some village there are $1000$ inhabitants. Every day, each of them shares the news they learned yesterday with all their friends. It is known that any news becomes known to all residents of the village. Prove that it is possible to select $90$ residents so that if you tell all of them at the same time some news, then in $10$ days it will become known to all residents of the village.

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

For a finite set $ X$ of positive integers, let $ \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $ S$ of positive integers for which $ \Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $ T$ of positive integers for which $ S \subset T$ and $ \Sigma(S) \equal{} \frac{\pi}{2}.$ [i]Kevin Buzzard, United Kingdom[/i]