Found problems: 18
2015 District Olympiad, 1
Consider the parallelogram $ ABCD, $ whose diagonals intersect at $ O. $ The bisector of the angle $ \angle DAC $ and that of $ \angle DBC $ intersect each other at $ T. $ Moreover, $ \overrightarrow{TD} +\overrightarrow{TC} =\overrightarrow{TO} . $
Find the angles of the triangle $ ABT. $
2004 Germany Team Selection Test, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
2019 District Olympiad, 2
Let $H$ be the orthocenter of the acute triangle $ABC.$ In the plane of the triangle $ABC$ we consider a point $X$ such that the triangle $XAH$ is right and isosceles, having the hypotenuse $AH,$ and $B$ and $X$ are on each part of the line $AH.$ Prove that $\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB}$ if and only if $ \angle BAC=45^{\circ}.$
2000 Mongolian Mathematical Olympiad, Problem 2
Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define
$$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.
1991 All Soviet Union Mathematical Olympiad, 554
Do there exist $4$ vectors in the plane so that none is a multiple of another, but the sum of each pair is perpendicular to the sum of the other two? Do there exist $91$ non-zero vectors in the plane such that the sum of any $19$ is perpendicular to the sum of the others?
2023 Romania National Olympiad, 3
Let $n \geq 2$ be a natural number. We consider a $(2n - 1) \times (2n - 1)$ table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in $2n^2$ rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in $2n^2$ vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.
1999 Estonia National Olympiad, 4
For the given triangle $ABC$, prove that a point $X$ on the side $AB$ satisfies the condition $\overrightarrow{XA} \cdot\overrightarrow{XB} +\overrightarrow{XC} \cdot \overrightarrow{XC} = \overrightarrow{CA} \cdot \overrightarrow{CB} $, iff $X$ is the basepoint of the altitude or median of the triangle $ABC$.
2004 Germany Team Selection Test, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
2003 IMO Shortlist, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
VMEO I 2004, 4
In a quadrilateral $ABCD$ let $E$ be the intersection of the two diagonals, I the center of the parallelogram whose vertices are the midpoints of the four sides of the quadrilateral, and K the center of the parallelogram whose sides pass through the points. divide the four sides of the quadrilateral into three equal parts (see illustration ).
[img]https://cdn.artofproblemsolving.com/attachments/1/c/8f2617103edd8361b8deebbee13c6180fa848b.png[/img]
a) Prove that $\overrightarrow{EK} =\frac43 \overrightarrow{EI}$.
b) Prove that $$\lambda_A \overrightarrow{KA} +\lambda_B \overrightarrow{KB} + \lambda_C \overrightarrow{KC} + \lambda_D \overrightarrow{KD} = \overrightarrow{0}$$ , where
$$\lambda_A=1+\frac{S(ADB)}{S(ABCD)},\lambda_B=1+\frac{S(BCA)}{S(ABCD)},\lambda_C=1+\frac{S(CDB)}{S(ABCD)},\lambda_D=1+\frac{S(DAC)}{S(ABCD)}$$
, where $S$ is the area symbol.
2022 Harvard-MIT Mathematics Tournament, 10
On a board the following six vectors are written: $$(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1).$$ Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt2} (v + w)$ and $\frac{1}{\sqrt2} (v - w)$. After some number of moves, the sum of the six vectors on the board is $u$. Find, with proof, the maximum possible length of $u$.
1983 Polish MO Finals, 5
On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$. Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$.
1992 Tournament Of Towns, (331) 3
Let $O$ be the centre of a regular $n$-gon whose vertices are labelled $A_1$,$...$, $A_n$. Let $a_1>a_2>...>a_n>0$. Prove that the vector
$$a_1\overrightarrow{OA_1}+a_2\overrightarrow{OA_2}+...+a_n\overrightarrow{OA_n}$$
is not equal to the zero vector.
(D. Fomin, Alexey Kirichenko)
2000 Saint Petersburg Mathematical Olympiad, 11.2
Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality:
$$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$
[I]Proposed by A. Khrabrov[/i]
ICMC 5, 5
A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$.
[i]Proposed by Ethan Tan[/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 $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.
1999 Slovenia National Olympiad, Problem 2
Three unit vectors $a,b,c$ are given on the plane. Prove that one can choose the signs in the expression $x=\pm a\pm b\pm c$ so as to obtain a vector $x$ with $|x|\le\sqrt2$.
2011 Bogdan Stan, 2
Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $
[i]Ion Tecu[/i]