Found problems: 560
2001 Romania National Olympiad, 3
Let $n\in\mathbb{N}^*$ and $v_1,v_2,\ldots ,v_n$ be vectors in the plane with lengths less than or equal to $1$. Prove that there exists $\xi_1,\xi_2,\ldots ,\xi_n\in\{-1,1\}$ such that
\[ | \xi_1v_1+\xi_2v_2+\ldots +\xi_nv_n|\le\sqrt{2}\]
1981 Spain Mathematical Olympiad, 6
Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, .
Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?
1985 Greece National Olympiad, 1
Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$
where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively
2002 China Western Mathematical Olympiad, 4
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}}$.
2005 Miklós Schweitzer, 10
Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.
1974 All Soviet Union Mathematical Olympiad, 193
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] $2$.
2011 JBMO Shortlist, 6
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
1995 Taiwan National Olympiad, 6
Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.
2004 All-Russian Olympiad, 2
Prove that there is no finite set which contains more than $ 2N,$ with $ N > 3,$ pairwise non-collinear vectors of the plane, and to which the following two characteristics apply:
1) for $ N$ arbitrary vectors from this set there are always further $ N\minus{}1$ vectors from this set so that the sum of these is $ 2N\minus{}1$ vectors is equal to the zero-vector;
2) for $ N$ arbitrary vectors from this set there are always further $ N$ vectors from this set so that the sum of these is $ 2N$ vectors is equal to the zero-vector.
2007 Romania National Olympiad, 1
Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.
1995 China National Olympiad, 3
Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow:
$x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$;
$ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $
$i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$.
Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.
2010 Romanian Master of Mathematics, 5
Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$, there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$, where each $T_i$ is distance $1$ away from $T_{i+1}$. For such a set $K$, we define the set of vectors
\[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\]
What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane?
[i]Grigory Chelnokov, Russia[/i]
1988 India National Olympiad, 8
A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.
2010 Iran MO (3rd Round), 4
a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms:
i-$\{0\}$.
ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$.
iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$)
b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)
2014 Contests, 3
Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.
2005 China Team Selection Test, 3
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?
JBMO Geometry Collection, 2011
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
MathLinks Contest 7th, 1.3
We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least
\[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}.
\]
2010 IFYM, Sozopol, 2
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 $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.
1977 IMO Longlists, 33
A circle $K$ centered at $(0,0)$ is given. Prove that for every vector $(a_1,a_2)$ there is a positive integer $n$ such that the circle $K$ translated by the vector $n(a_1,a_2)$ contains a lattice point (i.e., a point both of whose coordinates are integers).
2014 Miklós Schweitzer, 6
Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.
2004 Romania National Olympiad, 1
On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that:
(a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$;
(b) $P,A,Q$ are collinear.
[i]Virginia Tica, Vasile Tica[/i]
1966 IMO Longlists, 22
Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$
Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $%
P^{\prime }.$
2002 China Team Selection Test, 2
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
2011 Putnam, B4
In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$
Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$