Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a right octagon with center $O$ and $\lambda_1$,$\lambda_2$, $\lambda_3$, $\lambda_4$ be some rational numbers for which: $\lambda_1 \overrightarrow{OA_1}+\lambda_2 \overrightarrow{OA_2}+\lambda_3 \overrightarrow{OA_3}+\lambda_4 \overrightarrow{OA_4} =\overrightarrow{o}$. Prove that $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.

In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)

On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.

[b]a)[/b] Let $ AA',BB',CC' $ be the altitudes of a triangle $ ABC. $ Prove that $$ \frac{BC}{AA'}\cdot \overrightarrow{AA'} +\frac{AC}{BB'}\cdot \overrightarrow{BB'} +\frac{AB}{CC'}\cdot \overrightarrow{CC'} =0. $$ [b]b)[/b] The sum of the vectors that are perpendicular to the sides of a convex polygon and have equal lengths as those sides, respectively, is $ 0. $

Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that \[ \sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, . \]

Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$. a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$; b) Show that $MW$ is perpendicular to $AY$.

A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$. [asy]import math; unitsize(5mm); defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7)); pair O=(0,0); pair A=(0,sqrt(17)); pair B=(sqrt(17),0); pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75)); pair D=(xpart(C),8); pair E=(8,ypart(C)); draw(O--(0,8)); draw(O--(8,0)); draw(O--C); draw(A--C--B); draw(D--C--E); label("$\sqrt{17}$",(0,2),W); label("$\sqrt{17}$",(2,0),S); label("cut",midpoint(A--C),NNW); label("cut",midpoint(B--C),ESE); label("fold",midpoint(C--D),W); label("fold",midpoint(C--E),S); label("$30^\circ$",shift(-0.6,-0.6)*C,WSW); label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]

Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2=\sqrt{11}$. Find $\sum^4_{k=1}(CE_k)^2$.

We call a set of $6$ points in the plane [i]splittable[/i] if we if can denote its elements by $A,B,C,D,E$ and $F$ in such a way that $\triangle ABC$ and $\triangle DEF$ have the same centroid. [list=a] [*]Construct a splittable set. [*]Show that any set of $7$ points has a subset of $6$ points which is [i]not[/i] splittable. [/list]

Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.

In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum \[ \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? \] $ \textbf{(A) }1011 \qquad \textbf{(B) }1012 \qquad \textbf{(C) }2023 \qquad \textbf{(D) }2024 \qquad \textbf{(E) }2025 \qquad $

Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

Parallelogram $ ABCD$ has area $ 1,\!000,\!000$. Vertex $ A$ is at $ (0,0)$ and all other vertices are in the first quadrant. Vertices $ B$ and $ D$ are lattice points on the lines $ y\equal{}x$ and $ y\equal{}kx$ for some integer $ k>1$, respectively. How many such parallelograms are there? $ \textbf{(A)}\ 49\qquad \textbf{(B)}\ 720\qquad \textbf{(C)}\ 784\qquad \textbf{(D)}\ 2009\qquad \textbf{(E)}\ 2048$

Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$. On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC. The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$ Prove that \[AM\perp PQ.\] [b]Author: Marin Bancoș Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]

Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: [i]for every[/i] $k, k\le n$[i], the length of the sum of the first[/i] $k$ [i]vectors is less than[/i] $2$.

Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

Participants to an olympiad worked on $n$ problems. Each problem was worth a [color=#FF0000]positive [/color]integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants.

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

Let $n\ge 2$ be a positive integer. Let $x_1, x_2, \dots, x_n$ be non-zero real numbers satisfying the equation \[\left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)\dots\left(x_n+\frac{1}{x_1}\right)=\left(x_1^2+\frac{1}{x_2^2}\right)\left(x_2^2+\frac{1}{x_3^2}\right)\dots\left(x_n^2+\frac{1}{x_1^2}\right).\] Find all possible values of $x_1, x_2, \dots, x_n$. [i]Proposed by Victor Domínguez[/i]

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 $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$, respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$, $C$ is the second point of intersection of $AB$ and $\Gamma'$, and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$.

Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.