Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y +\frac{349}{4}= 0$. Find the sum of all possible values of $x$.

Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.

An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus? $ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$ [asy]unitsize(1.4cm); defaultpen(linewidth(.8pt)); dotfactor=3; real r1=1.0, r2=1.8; pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0]; pair[] points={X,O,Y,Z}; filldraw(Circle(O,r2),mediumgray,black); filldraw(Circle(O,r1),white,black); dot(points); draw(X--Y--O--cycle--Z); label("$O$",O,SSW,fontsize(10pt)); label("$Z$",Z,SW,fontsize(10pt)); label("$Y$",Y,N,fontsize(10pt)); label("$X$",X,NE,fontsize(10pt)); defaultpen(fontsize(8pt)); label("$c$",midpoint(O--Z),W); label("$d$",midpoint(Z--Y),W); label("$e$",midpoint(X--Y),NE); label("$a$",midpoint(X--Z),N); label("$b$",midpoint(O--X),SE);[/asy]

A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$. Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$. a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$. b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). How many $4$-digit numbers include $4$ consecutive digits? [i](9 and 0 are not considered consecutive digits.)[/i] $\text{(A) }18\qquad\text{(B) }24\qquad\text{(C) }144\qquad\text{(D) }162\qquad\text{(E) }168$

Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?

There are $n\geq2$ weights such that each weighs a positive integer less than $n$ and their total weights is less than $2n$. Prove that there is a subset of these weights such that their total weights is equal to $n$.

If two faces of a dice have a common edge, the two faces are called adjacent faces. In how many ways can we construct a dice with six faces such that any two consecutive numbers lie on two adjacent faces? $\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \text{None}$

The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if: a) it is hockey (draws allowed)? b) it is volleyball (no draws)?

In $4$-dimensional space, a set of $1 \times 2 \times 3 \times 4$ bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks: [list]i) $2 \times 5 \times 7 \times 12$ ii) $5 \times 5 \times 10 \times 12$ iii) $6 \times 6 \times 6 \times 6$.[/list]

Alice, Bob, and Catherine decide to have a race. Alice runs at a speed of $3$ feet per second, and Bob runs at a speed of $5$ feet per second. In the end, Bob finishes the same amount of time before Catherine as Catherine finishes before Alice. What was Catherine's speed, in feet per second? $\textbf{(A) }\dfrac{15}4\qquad\textbf{(B) }4\qquad\textbf{(C) }\dfrac{17}4\qquad\textbf{(D) }\dfrac92\qquad\textbf{(E) }\text{impossible to determine}$

It is known about the function $f : R \to R$ that $\bullet$ $f(x) > f(y)$ for all real $x > y$ $\bullet$ $f(x) > x$ for all real $x$ $\bullet$ $f(2x - f (x)) = x$ for all real $x$. Prove that $f(x) = x + f(0)$ for all real numbers $x$.

Triangle $ABC$ has points $D$,$E$,$F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$, and $AD$ and $AE$ trisect angle $BAF$. If $\angle BAF = 60^{\circ}$, $\frac{EF}{EC}=\frac{2}{3}$, and $\frac{AE}{AC} = 2$, find $\angle BAC$. [i]Individual #5[/i]

Let $ n$ be a nonzero positive integer. Find $ n$ such that there exists a permutation $ \sigma \in S_{n}$ such that \[ \left| \{ |\sigma(k) \minus{} k| \ : \ k \in \overline{1, n} \}\right | = n.\]

A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?

On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]

In acute-angled triangle $ ABC$, the points $ D,E,F$ are on sides $ BC,CA,AB$, respectively such that $ \angle AFE \equal{} \angle BFD, \angle FDB \equal{} \angle EDC, \angle DEC \equal{} \angle FEA$. Prove that $ AD$ is perpendicular to $ BC$.

a) A bus network is organized so that: 1) one can reach any stop from any other stop without changing buses; 2) every pair of routes has a single stop at which one can change buses; 3) each route has exactly three stops? How many bus routes are there? It is assumed that there are at least two routes. b) A town has $57$ bus routes. How many stops does each route have if it is known that 1) one can reach any stop from any other stop without changing buses; 2) for every pair of routes there is a single stop where one can change buses; 3) each route has three or more stops?

$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.

There is a WORD game with the following rules. There are finite number of relations $U_{i}\longrightarrow V_{i}$($U_{i},V_{i}$ are words). There is are two words $A,B$. We start from $A$, and we want to reach to $B$. At each step we can change one subword $U_{i}$ to $V_{i}$. Prove that there does not exist an algorithm that picks up $A,B$ and $U_{i}$'s,$V_{i}$'s and decides whether we can reach from $A$ to $B$ or not.

Let $ n$ be a given positive integer. Show that we can choose numbers $ c_k\in\{\minus{}1,1\}$ ($ i\le k\le n$) such that \[ 0\le\sum_{k\equal{}1}^nc_k\cdot k^2\le4.\]