Found problems: 560
Let $ Z_1,\,Z_2\dots,\,Z_n$ be $ d$-dimensional independent random (column) vectors with standard normal distribution, $ n \minus{} 1 > d$. Furthermore let
\[ \overline Z \equal{} \frac {1}{n}\sum_{i \equal{} 1}^n Z_i,\quad S_n \equal{} \frac {1}{n \minus{} 1}\sum_{i \equal{} 1}^n(Z_i \minus{} \overline Z)(Z_i \minus{} \overline Z)^\top\]
be the sample mean and corrected empirical covariance matrix. Consider the standardized samples $ Y_i \equal{} S_n^{ \minus{} 1/2}(Z_i \minus{} \overline Z)$, $ i \equal{} 1,2,\dots,n$. Show that
\[ \frac {E|Y_1 \minus{} Y_2|}{E|Z_1 \minus{} Z_2|} > 1,\]
and that the ratio does not depend on $ d$, only on $ n$.
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
Let $H{}$ be the orthocenter of the triangle $ABC{}$ and $X{}$ be the midpoint of the side $BC.$ The perpendicular at $H{}$ to $HX{}$ intersects the sides $(AB)$ and $(AC)$ at $Y{}$ and $Z{}$ respectively. Let $O{}$ be the circumcenter of $ABC{}$ and $O'$ be the circumcenter of $BHC.$ [list=a]
[*]Prove that $HY=HZ.$
[*]Prove that $\overrightarrow{AY}+\overrightarrow{AZ}=2\overrightarrow{OO'}.$
[/list]
Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$.
(translated by L. Erdős)
In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.
The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$.
$ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t.
$ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.
Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for
every column vector $u\in \mathbb{R}^{3}$. Prove that:
a) $A^{T}=-A$.
b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$,
where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$.
Let $A_1,A_2,\ldots,A_{2017}$ be the vertices of a regular polygon with $2017$ sides.Prove that there exists a point $P$ in the plane of the polygon such that the vector
$$\sum_{k=1}^{2017}k\frac{\overrightarrow{PA}_k}{\left\lVert\overrightarrow{PA}_k\right\rVert^5}$$
is the zero vector.
(The notation $\left\lVert\overrightarrow{XY}\right\rVert$ represents the length of the vector $\overrightarrow{XY}$.)
Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$
Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$.
Show that the points $B$, $X$, $H$, $Y$ lie on one circle.
Let $P,Q,R,S$ be the midpoints of the sides $BC,CD,DA,AB$ of a convex quadrilateral, respectively. Prove that
\[4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)\]
Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows:
$+$ $0$ $a$ $b$ $c$
$0$ $0$ $a$ $b$ $c$
$a$ $a$ $0$ $c$ $b$
$b$ $b$ $c$ $0$ $a$
$c$ $c$ $b$ $a$ $0$
a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$.
b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.
$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.
Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$, where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$. Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$. Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$, where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$.
Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side.
[i]Note[/i]. If the pentagon is labeled $ ABCDE$, the adjacent vertices of $ A$ are $ B$ and $ E$, the ones of $ B$ are $ A$ and $ C$ etc.
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
a) Let $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3}$ and $A=(a_{ij})\in \mathcal{M}_3$ Prove that $\det A=0$.
b) Let $z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3$ such that $|z_1|=|z_2|=\ldots=|z_{n+3}|$ and $\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3})$. If $b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n}$ and $B=(b_{ij})\in \mathcal{M}_n$, prove that $\det B=0$.
Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.
The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$, $B'$, $C'$, and $D'$, where $AA' = k/h_a$, $BB' = k/h_b$, $CC' = k/h_c$, and $DD' = k/h_d$. Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$, etc. Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$.
Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$
is independent of positions of $K,L,M,N$, (i.e. is constant )
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
\[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}.
\]
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]
Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector?
(Problem from Leningrad)
Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$.
[i]J. Szucs[/i]