A trapezoid $ABCD$ is bicentral. The vertex $A$, the incenter $I$, the circumcircle $\omega$ and its center $O$ are given and the trapezoid is erased. Restore it using only a ruler.

Given pairwise coprime natural numbers $ x $, $ y $, $ z $, $ t $ such that $ xy + yz + zt = xt $. Prove that the sum of the squares of some two of these numbers is twice the sum of the squares of the two remaining.

Point $C$ lies inside the right angle $AOB$. Prove that the perimeter of triangle $ABC$ is greater than $2 OC$.

For all positive integers $n$, let $p(n)$ be the number of non-decreasing sequences of positive integers such that for each sequence, the sum of all terms of the sequence is equal to $n$. Prove that \[\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.\]

Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.

A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine $ a_{1998}$.

Let ABCD be a cyclic quadrilateral in which AB = BC and AD =CD. Point M is on the small arc CD of the circle circumscribed to the quadrilateral. The lines BM and CD intersect at point P, and the lines AM and BD intersect at point Q. Prove that PQ is parralel to AC.

Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle. [i]Proposed by Karthik Vedula[/i]

There are at least $3$ subway stations in a city. In this city, there exists a route that passes through more than $L$ subway stations, without revisiting. Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A. Prove that at least one of the two holds. $\text{(i)}$. There exists three subway stations $A$, $B$, $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$. $\text{(ii)}$. There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.

Find all integers $x$ and $y$ such that $2^x+1=y^2$

Four forces acting on a body are in equilibrium. Prove that, if their lines of action are mutually skew, they are rulings of a hyperboloid.

Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.

If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.

The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.

Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$  at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects  $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$. [i]Proposed by Ali Zamani [/i]

Find the minimum possible value of the largest of $xy$, $1-x-y+xy$, and $x+y-2xy$ if $0\leq x \leq y \leq 1$.

Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$. Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel. (Here, the points $P$ and $Q$ are [i]isogonal conjugates[/i] with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.) [i]Proposed by Sammy Luo[/i]

Suppose $a_0,a_1,\ldots, a_{2018}$ are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$.

For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)

Consider an acute-angled triangle $\triangle ABC$ ($AC>AB$) with its orthocenter $H$ and circumcircle $\Gamma$.Points $M$,$P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$. Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP=\angle HNM$ ($B$ and $J$ lie one the same side of $\overline{AH}$) and circle $\omega_1$, passing through $K,H$, and $J$, and circle $\omega_2$ passing through $K,M$, and $N$, are externally tangent to each other. Prove that the common external tangents of $\omega_1$ and $\omega_2$ meet on the line $\overline{NH}$. [i]Proposed by Alireza Dadgarnia[/i]

A numerist has $n$ eurodollars and distributes them to two bank accounts $A, B$ in Germany and Switzerland, whereby the Eurodollars cannot be split into smaller monetary units due to the lack of a suitable name. In order to hide all money from the tax authorities if necessary, he would like to be able to move all of his money to account $B$. Due to the immense bureaucracy, money is only allowed to be moved between two accounts if the deposited amount in one account is double. Of course, he can carry out several such transfers in a row. Show that the number of ways to initially distribute the money appropriately is a power of two.

In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$.

Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$

$S=\sin{64x}+\sin{65x}$ and $C=\cos{64x}+\cos{65x}$ are both rational for some $x$. Prove, that for one of these sums both summands are rational too.

Given a $\triangle ABC$ and the altitude $CH$ ($H$ lies on the segment $AB$) and let $M$ be the midpoint of $AC$. Prove that if the circumcircle of $\triangle ABC$, $k$ and the circumcircle of $\triangle MHC$, $k_{1}$ touch, then the radius of $k$ is twice the radius of $k_{1}$.