Found problems: 623
Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations?
i)$k$ is a prime number greater than $2$;
ii) $k$ is odd;
iii) $k$ is even.
Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$.
[i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]
In the group of ten persons, each person is asked to write the sum of the ages of all the other nine persons. Of all ten sums form the nine-element set $\{ 82, 83,84,85,87,89,90,91,92 \}$, find the individual ages of the persons, assuming them to be whole numbers.
Several points are denoted on a white piece of paper. The distance between each two of the points is greater than $24$. A drop of ink was sprinkled over the paper covering an area smaller than $\pi$. Prove that there exists a vector $\overrightarrow v$ with $\overrightarrow v<1$, such that after translating all of the points by $v$ none of them is covered in ink.
There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.
Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy
\begin{align*}
& \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\
& \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\
& \;\qquad \vdots \\
& \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\
\end{align*}
For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?
Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if
\[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\]
for every prime number $p$ and every real number $y.$
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.
Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.
A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times 8=32$.
Find the greatest possible value of $n$, such that any four-element set with elements less than or equal to $n$ is good.
[i]Proposed by Victor and Isaías de la Fuente[/i]
Let’s consider three pairwise non-parallel straight constant lines in the plane. Three points are moving along these lines with different nonzero velocities, one on each line (we consider the movement to have taken place for infinite time and continue infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some “zero” moment in such a way that the points never were, are or will be collinear?
[b]a)[/b] Prove that, for any point in the interior of a triangle, there are two points on the sides of this triangle such that the resultant of the vectors from the interior point those two points is the vector $ 0. $
[b]b)[/b] Prove that, for any point in the interior of a triangle, there are three points on the sides of this triangle such that the resultant of the vectors from the interior point those three points is the vector $ 0. $
For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$
(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.
(b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$
If $a, b, c$ are non-negative real numbers with $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 1$, prove that:
\[ \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2\]
Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !
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.
Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.
Calculate the sum of vector products $[[x, y], z] + [[y, z], x] + [[z, x], y]$
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.
$ ABC$ and $ A'B'C'$ are two triangles in the same plane such that the lines $ AA',BB',CC'$ are mutually parallel. Let $ [ABC]$ denotes the area of triangle $ ABC$ with an appropriate $ \pm$ sign, etc.; prove that
\[ 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].\]
Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$-module of rank $2m+1$, equipped with a perfect symmetric $\mathbb{Z}/p^N\mathbb{Z}$-bilinear form
\[(\, ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}.\]
Here ``perfect'' means that the induced map
\[\Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}), \quad x \mapsto (x,\cdot)\]
is an isomorphism. Find the cardinality of the set
\[\{x \in \Lambda: (x,x)=0\},\]
expressed in terms of $p,m,N$.
Let $n$ be a positive integer. Suppose we are given $2^n+1$ distinct sets, each containing finitely many objects. Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category. We define the [i]symmetric difference[/i] of two sets as the set of objects belonging to exactly one of the two sets. Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set.
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
Given a point $X$ and $n$ vectors $\overrightarrow{x_i}$ with sum zero in the plane. For each permutation of the vectors we form a set of $n$ points, by starting at $X$ and adding the vectors in order. For example, with the original ordering we get $X_1$ such that $XX_1 = \overrightarrow{x_1}, X_2$ such that $X_1X_2 = \overrightarrow{x_2}$ and so on. Show that for some permutation we can find two points $Y, Z$ with angle $\angle YXZ = 60^o $, so that all the points lie inside or on the triangle $XYZ$.
Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.