We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

If $ a\equal{}\minus{}2$, the largest number in the set $ \left \{ \minus{}3a,4a,\frac{24}{a},a^2,1 \right \}$ is \[ \textbf{(A)}\ \minus{}3a \qquad \textbf{(B)}\ 4a \qquad \textbf{(C)}\ \frac{24}{a} \qquad \textbf{(D)}\ a^2 \qquad \textbf{(E)}\ 1 \]

Consider $ \triangle ABC$ and points $ M \in (AB)$, $ N \in (BC)$, $ P \in (CA)$, $ R \in (MN)$, $ S \in (NP)$, $ T \in (PM)$ such that $ \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k$ and $ \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k$ for some $ k \in (0, 1)$. Prove that $ \triangle STR \sim \triangle ABC$ and, furthermore, determine $ k$ for which the minimum of $ [STR]$ is attained.

Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.

Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.

Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$

Danielle draws a point $O$ on the plane and a set of points $\mathcal P = \{P_0, P_1, \ldots , P_{2022}\}$ such that $$\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,$$where the angles are measured counterclockwise and for $0 \leq n \leq 2022$, $OP_n = r^n$, where $r > 1$ is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points $\{A_0, A_1, \ldots , A_n\}$ in the plane, it is built a new set of points $\{B_0, B_1, \ldots , B_{n-1}\}$ such that $A_kA_{k+1}B_k$ is an equilateral triangle oriented clockwise for $0 \leq k \leq n - 1$. After carrying out the process $2022$ times from the set $P$, Danielle obtains a single point $X$. If the distance from $X$ to point $O$ is $d$, show that $$(r-1)^{2022} \leq d \leq (r+1)^{2022}.$$

Let $ABC$ be a triangle. Let $P, Q, R$ be the midpoints of $BC, CA, AB$ respectively. Let $U, V, W$ be the midpoints of $QR, RP, PQ$ respectively. Let $x=AU, y=BV, z=CW$. Prove that there exist a triangle with sides $x, y, z$.

Let $ O$ be an interior point of the triangle $ ABC$. Prove that there exist positive integers $ p,q$ and $ r$ such that \[ |p\cdot\overrightarrow{OA} \plus{} q\cdot\overrightarrow{OB} \plus{} r\cdot\overrightarrow{OC}|<\frac{1}{2007}\]

$20$ points with no three collinear are given. How many obtuse triangles can be formed by these points? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 2{{10}\choose{3}} \qquad \textbf{(D)}\ 3{{10}\choose{3}} \qquad \textbf{(E)}\ {{20}\choose{3}}$

Let $ A$ be a family of proper closed subspaces of the Hilbert space $ H\equal{}l^2$ totally ordered with respect to inclusion (that is , if $ L_1,L_2 \in A$, then either $ L_1\subset L_2$ or $ L_2\subset L_1$). Prove that there exists a vector $ x \in H$ not contaied in any of the subspaces $ L$ belonging to $ A$. [i]B. Szokefalvi Nagy[/i]

Given two vectors $\overrightarrow a$, $\overrightarrow b$, find the range of possible values of $\|\overrightarrow a - 2 \overrightarrow b\|$ where $\|\overrightarrow v\|$ denotes the magnitude of a vector $\overrightarrow v$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 1)[/i]