Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1 + ax_2 + x_3 = b $$ $$x_2 + ax_3 + x_4 = b $$ $$x_3 + ax_4 + x_1 = b $$ $$x_4 + ax_1 + x_2 = b$$ Here $a$ and $b$ are real numbers (case distinction!).

Let $P_{n}$ denote the number of paths in the coordinate plane traveling from $(0, 0)$ to $(n, 0)$ with three kinds of moves: [i]upstep[/i] $u = [1, 1]$, [i]downstep[/i] $d = [1,-1]$, and [i]flatstep[/i] $f = [1, 0]$ with the path always staying above the line $y = 0.$ Let $C_{n}= \frac{1}{n+1}\binom{2n}{n}$ be the $n^{th}$ Catalan number. Prove that $P_{n}= \sum_{i = 0}^\infty \binom{n}{2i}C_{i}$ and $C_{n}= \sum_{i = 0}^{2n}(-1)^{i}\binom{2n}{i}P_{2n-i}.$ [hide="Solution to Part 1"] Let a path string, $S_{k}$, denote a string of $u, d, f$ corresponding to upsteps, downsteps, and flatsteps of length $k$ which successfully travels from $(0, 0)$ to $(n, 0)$ without passing below $y = 0.$ Also, let each entry of a path string be a slot. Lastly, denote $u_{k}, d_{k}, f_{k}$ to be the number of upsteps, downsteps, and flatsteps, respectively, in $S_{k}.$ Note that in our situation, all such path strings are in the form $S_{n},$ so all our path strings have $n$ slots. Since the starting and ending $y$ values are the same, the number of upsteps must equal the number of downsteps. Let us observe the case when there are $2k$ downsteps and upsteps totally. Thus, there are $\binom{n}{2k}$ ways to choose the slots in which the upsteps and the downsteps appear. Now, we must arrange the downsteps and upsteps in such a way that $d_{n}= u_{n}$ and a greater number of upsteps preceed downsteps, as the path is always above $y = 0$. Note that a bijection exists between this and the number of ways to binary bracket $k$ letters. The number of binary brackets of $k$ letters is just the $k^{th}$ Catalan number. We then place the flatsteps in the rest of the slots. Thus, there are a total of $\sum_{k = 0}^\infty \binom{n}{2k}C_{k}$ ways to get an $S_{n}.$ [/hide]

We call $A_{1},A_{2},A_{3}$ [i]mangool[/i] iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$. To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$. Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. [i]Brazitikos Silouanos, Greece[/i]

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

As everyone knows, the people of [i]Plane Land[/i] love Planimetrics. Therefore, they imagine their country as completely planar, every city in the country as a geometric point and every road as the line segment connecting two points. Additionally to the existing cities, it is possible to build [i]roundabouts[/i], i.e. points in the road network from where at least two roads emanate. All road crossings or junctions are build as roundabouts. Via this route network, every two cities should be connected by a sequence of roads and possibly roundabouts. In Plane Land, the length of a road is taken as the geometric length of the corresponding line segment. The ingenious road engineer Armin Asphalt presents a new road map, of which it is known that there is no road network with a smaller total length of all roads. Moreover, there is no road map with the same total length of all roads and fewer roundabouts. Prove that in the road map of Armin Asphalt, at most three roads emanate from each city, and exactly three from each roundabout.

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

The altitudes of a triangles are $12$, $15$, and $20$. The largest angle in this triangle is $\textbf{(A) }72^\circ\qquad\textbf{(B) }75^\circ\qquad\textbf{(C) }90^\circ\qquad\textbf{(D) }108^\circ\qquad\textbf{(E) }120^\circ$

The unit of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed with the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to only make one purchase, what is the maximum possible number of screws he can buy?

Let $a<b<c$ be three positive integers. Prove that among any $2c$ consecutive positive integers there exist three different numbers $x,y,z$ such that $abc$ divides $xyz$.

On the coordinate plane is given the square with vertices $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$. For every $n\in\mathbb N$, point $T_{n+4}$ is defined as the midpoint of the segment $T_nT_{n+1}$. Determine the coordinates of the limit point of $T_n$ as $n\to\infty$, if it exists.

There is scales on the teacher's table. There is a set of weighs on the scales, and there are some pupils' names (may be more than one) on the every weigh. A pupil entering the classroom moves all the weight with his name to another side of the scales. Prove that you can let in such a subset of the pupils, that the scales will change its position.

Given any integer $n\geq 3$. A finite series is called $n$-series if it satisfies the following two conditions $1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$ $2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$ How many $n$-series are there $?$

A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.

$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.

Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. Find the maximum value of $x^4y+xy^4$.