Found problems: 623
Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.
Let $S=\{a_1,a_2,\ldots,a_n\}$ of nenul vectors in a plane. Show that $S{}$ can be partitioned in nenul subsets $B_1, B_2,\ldots, B_m$ with the properties:
1) each vector from $S{}$ is part of only on subset;
2) if $a_i\in B_j$ then the angle between vectors $a_i$ and $c_j$, which is the sum of all vectors from $B_j$ is not greater than $\frac{\pi}{2}$;
3) if $i\neq j$ then the angle between vectors $c_i$ and $c_j$, which is the sum of all vectors from $B_i$ and $B_j$, respectively, is greater than $\frac{\pi}{2}$.
What are the possible values of $m$?
What is the size of the largest subset $S'$ of $S = \{2^x3^y5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$?
Find the parallel displacement of a vector pointing north at Leningrad (latitude $60^{\circ}$) from west to east along a closed parallel.
Let $ABCDEF$ be a convex hexagon and denote $U$,$V$,$W$,$X$,$Y$ and $Z$ be the midpoint of $AB$,$BC$,$CD$,$DE$,$EF$ and $FA$ respectively.
Prove that the length of $UX$,$VY$,$WZ$ can be the length of each sides of some triangle.
Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$.
( for example the diffrence of this two row is only in one index
110
100)
[i]Edited by Myth[/i]
We are given $30$ non-zero vectors in $3$ dimensional space.
Prove that among these there are two such that the angle between them is less than $45^o$.
Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$.
Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$. $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$. Let $D$ be the extouch at $BC$. Show that $AD$ is perpendicular to $B_2C_2$
In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$
Let $a_j$, $b_j$, $c_j$ be integers for $1 \le j \le N$. Assume for each $j$, at least one of $a_j$, $b_j$, $c_j$ is odd. Show that there exists integers $r, s, t$ such that $ra_j+sb_j+tc_j$ is odd for at least $\tfrac{4N}{7}$ values of $j$, $1 \le j \le N$.
Consider the plane vectors $\overrightarrow{OA_1},\overrightarrow{OA_2},\dots ,\overrightarrow{OA_n}$ with $n\geq 3$. Suppose that the inequality $$\big|\overrightarrow{OA_1}+\overrightarrow{OA_2}+\dots +\overrightarrow{OA_n}\big|\geq \big|\pm\overrightarrow{OA_1}\pm\overrightarrow{OA_2}\pm\dots \pm\overrightarrow{OA_n}\big|$$ takes place for all choiches of the $\pm$ signs. Show that there exists a line $\ell$ through $O$ such that all points $A_1,A_2,\dots ,A_n$ are all on one side of $\ell$.
[i]Cristi Săvescu[/i]
A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$
$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that
\[
\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.
\]
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$
Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.
here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad
assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n.
if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n},
we move P on the plane according to the S in this form: at first we reflect P through the s1
( s1 means the edge which iys number is s1)then through s2 and so on like the figure below.
a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane
b)prove that the sequence in a) isn't periodic.
c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?
Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.
Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist.
[i]L. Lovasz[/i]
$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$
Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$
Let $ M,N,P $ on the sides $ AB,BC,CA, $ respectively, of a triangle $ ABC $ such that $ AM=BN=CP $ and such that
$$ AB\cdot \overrightarrow{AT} +BC\cdot \overrightarrow{BT} +CA\cdot \overrightarrow{CT} =0, $$
where $ T $ is the centroid of $ MNP. $ Prove that $ ABC $ is equilateral.
The angle bisector of the acute angle formed at the origin by the graphs of the lines $y=x$ and $y=3x$ has equation $y=kx$. What is $k$?
$\textbf{(A)} \: \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \: \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \: \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \: 2\qquad \textbf{(E)} \: \frac{2+\sqrt{5}}{2}$