Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if $$(u+ix)(v+iy)(w+iz)=i?$$

Solve the inequation $\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$.

In Wonderland there are at least $5$ towns. Some towns are connected directly by roads or railways. Every town is connected to at least one other town and for any four towns there exists some direct connection between at least three pairs of towns among those four. When entering the public transportation network of this land, the traveller must insert one gold coin into a machine, which lets him use a direct connection to go to the next town. But if the traveller continues travelling from some town with the same method of transportation that took him there, and he has paid a gold coin to get to this town, then going to the next town does not cost anything, but instead the traveller gains the coin he last used back. In other cases he must pay just like when starting travelling. Prove that it is possible to get from any town to any other town by using at most $2$ gold coins.

Determine all natural numbers $n \ge 2$ with the property that there are two permutations $(a_1, a_2,... , a_n) $ and $(b_1, b_2,... , b_n)$ of the numbers $1, 2,..., n$ such that $(a_1 + b_1, a_2 +b_2,..., a_n + b_n)$ are consecutive natural numbers. [i](Walther Janous)[/i]

Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions: (1)$a_1=1$ (2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$ what is the residue when we divide $f(2015)$ by $4$ ?

A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals 180 degrees.

Let $A_0A_1A_2A_3A_4A_5$ be a convex hexagon inscribed in a circle. Define the points $A_0'$, $A_2'$, $A_4'$ on the circle, such that \[ A_0A_0' \parallel A_2A_4, \quad A_2A_2' \parallel A_4A_0, \quad A_4A_4' \parallel A_2A_0 . \] Let the lines $A_0'A_3$ and $A_2A_4$ intersect in $A_3'$, the lines $A_2'A_5$ and $A_0A_4$ intersect in $A_5'$ and the lines $A_4'A_1$ and $A_0A_2$ intersect in $A_1'$. Prove that if the lines $A_0A_3$, $A_1A_4$ and $A_2A_5$ are concurrent then the lines $A_0A_3'$, $A_4A_1'$ and $A_2A_5'$ are also concurrent.

In ancient times there was a Celtic tribe consisting of several families. Many of these families were at odds with each other, so that their chiefs would not shake hands. At some point at the annual meeting of the chiefs they found it even impossible to assemble four or more of them in a circle with each of them being willing to shake his neighbour's hand. To emphasize the gravity of the situation, the Druid collected three pieces of gold from each family. The Druid then let all those chiefs shake hands who were willing to. For each handshake of two chiefs he paid each of them a piece of gold as a reward. Show that the number of pieces of gold collected by the Druid exceeds the number of pieces paid out by at least three.

Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively, and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively. Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$ meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that the triangles $BCM$ and $KLN$ are similar.

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.

Let $\ell$ be a positive integer. We say that a positive integer $k$ is [i]nice [/i] if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers. \\ \\ (Théo Lenoir)

Source: 2017 Canadian Open Math Challenge, Problem C4 ----- Let n be a positive integer and $S_n = \{1, 2, . . . , 2n - 1, 2n\}$. A [i]perfect pairing[/i] of $S_n$ is defined to be a partitioning of the $2n$ numbers into $n$ pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if $n = 4$, then a perfect pairing of $S_4$ is $(1, 8),(2, 7),(3, 6),(4, 5)$. It is not necessary for each pair to sum to the same perfect square. (a) Show that $S_8$ has at least one perfect pairing. (b) Show that $S_5$ does not have any perfect pairings. (c) Prove or disprove: there exists a positive integer $n$ for which $S_n$ has at least $2017$ different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)

Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on $\omega$.

The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$? $\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$

How many roots does equation $\sin x = \frac{x}{100}$ have?

Let $a$, $b$, $c$ and $d$ be positive real numbers with $a+b+c+d = 4$. Prove that $\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2$.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

Let $K$ be a closed subset of the closed unit ball in $\mathbb{R}^3$. Suppose there exists a family of chords $\Omega$ of the unit sphere $S^2$, with the following property: for every $X,Y\in S^2$, there exist $X',Y'\in S^2$, as close to $X$ and $Y$ correspondingly, as we want, such that $X'Y'\in \Omega$ and $X'Y'$ is disjoint from $K$. Verify that there exists a set $H\subset S^2$, such that $H$ is dense in the unit sphere $S^2$, and the chords connecting any two points of $H$ are disjoint from $K$. EDIT: The statement fixed. See post #4

If $ a$ and $ b$ are two unequal positive numbers, then: $ \textbf{(A)}\ \frac {2ab}{a \plus{} b} > \sqrt {ab} > \frac {a \plus{} b}{2} \qquad\textbf{(B)}\ \sqrt {ab} > \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2}$ $ \textbf{(C)}\ \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2} > \sqrt {ab} \qquad\textbf{(D)}\ \frac {a \plus{} b}{2} > \frac {2ab}{a \plus{} b} > \sqrt {ab}$ $ \textbf{(E)}\ \frac {a \plus{} b}{2} > \sqrt {ab} > \frac {2ab}{a \plus{} b}$

Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

Call a real number [i]amiable[/i] if it can be expressed in the form $a - b\sqrt{2}$, where $1 \le a, b \le 100$ are integers. Find the amiable number $x$ that minimizes $\left|x - \frac{1}{3}\right|$.

The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $$\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )$? $\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}$

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.

We say that a sequence $a_1,a_2,\cdots$ is [i]expansive[/i] if for all positive integers $j,\; i<j$ implies $|a_i-a_j|\ge \tfrac 1j$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.