Consider all non-empty subsets of the set $\{1,2\cdots,n\}$. For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as $S_n$. For example, \[S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3}\] [b](i)[/b] Show that $S_n=\frac1n+\left(1+\frac1n\right)S_{n-1}$. [b](ii)[/b] Hence or otherwise, deduce that $S_n=n$.

For any set $S$ of integers, let $f(S)$ denote the number of integers $k$ with $0 \le k < 2019$ such that there exist $s_1, s_2 \in S$ satisfying $s_1 - s_2 = k$. For any positive integer $m$, let $x_m$ be the minimum possible value of $f(S_1) + \dots + f(S_m)$ where $S_1, \dots, S_m$ are nonempty sets partitioning the positive integers. Let $M$ be the minimum of $x_1, x_2, \dots$, and let $N$ be the number of positive integers $m$ such that $x_m = M$. Compute $100M + N$. [i]Proposed by Ankan Bhattacharya[/i]

Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

In parallelogram $ABCD$, point $K$ is marked on diagonal $AC$. Circle $s_1$ passes through point $K$ and touches lines $AB$ and $AD$ ($s_1$ intersects the diagonal $AC$ for the second time on the segment $AK$). Circle $s_2$ passes through point $K$ and touches lines $CB$ and $CD$ ($s_2$ intersects for the second time diagonal $AC$ on segment $KC$). Prove that for all positions of the point $K$ on the diagonal $AC$, the straight lines connecting the centers of circles $ s_1$ and $s_2$, will be parallel to each other.

Show that if a holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ has the property that the modulus of any of its derivatives (of any order) is everywhere dominated by $ 1, $ then $ |f(z)|\le e^{|\text{Im} (z)|} , $ for all complex numbers $ z. $

In rectangular coodinate system $Oxy$, consider the line $y=3x$ and point $A(4,3)$. Find on the line $y=3x$, point $B\ne O$ such that the area of triangle $OBC$ is the minimum possible, where $C= AB\cap Ox$.

Let $o$, $r$, $g$, $t$, $n$, $i$, $z$, $e$, and $d$ be positive reals. Show that \[ \sqrt{(d+o+t+t+e+d)(o+r+z+i+n+g)} > \sqrt{ti} + \sqrt{go} + \sqrt[6]{orz}. \] when $d^2e \geq \tfrac{2}{1434}$. Proposed by [i]David Fox[/i]

Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$. Find the product $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$

Let $ABC$ be an acute-angled triangle such that $AB+BC=4AC$. Let $D$ in $AC$ such that $BD$ is angle bisector of $\angle ABC$. In the segment $BD$, points $P$ and $Q$ are marked such that $BP=2DQ$. The perpendicular line to $BD$, passing by $Q$, cuts the segments $AB$ and $BC$ in $X$ and $Y$, respectively. Let $L$ be the parallel line to $AC$ passing by $P$. The point $B$ is in a different half-plane(with respect to the line $L$) of the points $X$ and $Y$. An ant starts a run in the point $X$, goes to a point in the line $AC$, after that goes to a point in the line $L$, returns to a point in the line $AC$ and finishes in the point $Y$. Prove that the least length of the ant's run is equal to $4XY$.

Five distinct numbers are drawn successively and at random from the set $\{1, \cdots , n\}$. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than $\frac{6}{(n-2)^3}$

Let $i = \sqrt{-1}$. Define a sequence of complex numbers by $z_{1} = 0, z_{n+1} = z_{n}^{2}+i$ for $n \ge 1$. In the complex plane, how far from the origin is $z_{111}$? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ \sqrt{110}\qquad\textbf{(E)}\ \sqrt{2^{55}} $

Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$.

Given two relatively prime numbers $p>0$ and $q>0$. An integer $n$ is called "good" if we can represent it as $n = px + qy$ with nonnegative integers $x$ and $y$, and "bad" in the opposite case. a) Prove that there exist integer $c$ such that in a pair $\{n, c-n\}$ always one is "good" and one is "bad". b) How many there exist "bad" numbers?

In a triangle $ABC$, $\angle C = 60^o$, $D, E, F$ are points on the sides $BC, AB, AC$ respectively, and $M$ is the intersection point of $AD$ and $BF$. Suppose that $CDEF$ is a rhombus. Prove that $DF^2 = DM \cdot DA$

Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$.

Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds: \[\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}\]

Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?

In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, detemine the least $k$, such that, there exists a strategy that allows to identify the two false coins using the eletronic device, at most, $k$ times.

The bisectors of the angles $\sphericalangle CAB$ and $\sphericalangle BCA$ intersect the circumcircle of $ABC$ in $P$ and $Q$ respectively. These bisectors intersect each other in point $I$. Prove that $PQ \perp BI$.

Find all polynomials $p$ with real coefficients such that for all real $x$ $$(x-8)p(2x)=8(x-1)p(x).$$

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$? ${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $

Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?

You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.

Two circles $ \gamma_{1}$ and $ \gamma_{2}$ meet at points $ X$ and $ Y$. Consider the parallel through $ Y$ to the nearest common tangent of the circles. This parallel meets again $ \gamma_{1}$ and $ \gamma_{2}$ at $ A$, and $ B$ respectively. Let $ O$ be the center of the circle tangent to $ \gamma_{1},\gamma_{2}$ and the circle $ AXB$, situated outside $ \gamma_{1}$ and $ \gamma_{2}$ and inside the circle $ AXB.$ Prove that $ XO$ is the bisector line of the angle $ \angle{AXB}.$ [b]Radu Gologan[/b]