At least $ n - 1$ numbers are removed from the set $\{1, 2, \ldots, 2n - 1\}$ according to the following rules: (i) If $ a$ is removed, so is $ 2a$; (ii) If $ a$ and $ b$ are removed, so is $ a \plus{} b$. Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Let $A_1A_2... A_{2n + 1}$ be a convex polygon, $a_1 = A_1A_2$, $a_2 ​​= A_2A_3$, $...$, $a_{2n} = A_{2n}A_{2n + 1}$, $a_{2n + 1} = A_{2n + 1}A_1$. Denote by: $\alpha_i = \angle A_i$, $1 \le i \le 2n + 1$, $\alpha_{k + 2n + 1} = \alpha_k$, $k \ge 1$, $ \beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}$, $1 \le i \le 2n + 1$. Prove what if $$\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}$$ then a circle can be circumscribed around this polygon. Does the inverse statement hold a place?

The Fibonacci sequence is given by equalities $$F_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N$$. a) Prove that for every $m \ge 0$, the area of ​​the triangle $A_1A_2A_3$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$ is equal to $0.5$. b) Prove that for every $m \ge 0$ the quadrangle $A_1A_2A_4$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$, $A_4 (F_{m+7},F_{m+8})$ is a trapezoid, whose area is equal to $2.5$. c) Prove that the area of ​​the polygon $A_1A_2...A_n$ , $n \ge3$ with vertices does not depend on the choice of numbers $m \ge 0$, and find this area.

Which of the following is equivalent to $ \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}}$ when $ x < 0$? $ \textbf{(A) } \minus{} x \qquad \textbf{(B) } x \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \sqrt {\frac x2} \qquad \textbf{(E) } x\sqrt { \minus{} 1}$

Find real $a,b$ such that polynomial $P(x)=x^{n+1}+ax+b$ to be divisible by $(x-1)^2$. Then find the quotient $P(x):(x-1)^2 , n\in \mathbb{N}^*$

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

[b]3.[/b]Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$. Show that if, for any $k$, the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$, then every $H_k (k=1, \dots , n)$ is of finite index in $G$. [b](A. 15)[/b]

Let $ABC$ be an isosceles triangle ($AB=BC$) and $\ell$ be a ray from $B$. Points $P$ and $Q$ of $\ell$ lie inside the triangle in such a way that $\angle BAP=\angle QCA$. Prove that $\angle PAQ=\angle PCQ$.

Let $ABC$ be a scalene triangle. Let $D$ and $E$ be the points where the incircle touches sides $BC$ and $CA$, respectively. Let $K$ be the common point of line $BC$ and the bisector of the angle $\angle BAC$. Let $AD$ intersect $EK$ in $P$. Prove that $PC$ is perpendicular to $AK$.

Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$ . [hide=Original]$$f(x+f(y))+f(x+y)=2x+f(y)+y$$[/hide]

Two circles are intersecting in points $P$ and $Q$. Construct two points $A$ and $B$ on these circles so that $P\in AB$ and the product $AP.PB$ is maximal.

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.

In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.