Found problems: 85335
For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$
Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?
A [i]9-cube[/i] is a nine-dimensional hypercube (and hence has $2^9$ vertices, for example). How many five-dimensional faces does it have?
(An $n$ dimensional hypercube is defined to have vertices at each of the points $(a_1,a_2,\cdots ,a_n)$ with $a_i\in \{0,1\}$ for $1\le i\le n$)
[i]Proposed by Evan Chen[/i]
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another.
(For example, $23=9+8+6.)$
Let $ABC$ be a triangle and $D$ is a point inside of the triangle, such that $\angle DBC=60^{\circ}$ and $\angle DCB=\angle DAB=30^{\circ}$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. Prove that $\angle DMN=90^{\circ}$.
Seven points are selected randomly from $ S^1\subset\mathbb C$. What is the probability that origin is not contained in convex hull of these points?
Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.
Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.
Let $ABCD$ be a square, and let $M$ be the midpoint of side $BC$. Points $P$ and $Q$ lie on segment $AM$ such that $\angle BPD=\angle BQD=135^\circ$. Given that $AP<AQ$, compute $\tfrac{AQ}{AP}$.
Given a triangle $ABC$, in which the medians $BE$ and $CF$ are perpendicular. Let $M$ is the intersection point of the medians of this triangle, and $L$ is its Lemoine point (the intersection point of lines symmetrical to the medians with respect to the bisectors of the corresponding angles). Prove that $ML \perp BC$.
(Mykhailo Sydorenko)
The base-ten expressions of all the positive integers are written on an infinite ribbon without spacing: $1234567891011\ldots$. Then the ribbon is cut up into strips seven digits long. Prove that any seven digit integer will:
(a) appear on at least one of the strips; [i](3 points)[/i]
(b) appear on an infinite number of strips. [i](1 point)[/i]
Let $a,b,c,n$ be positive integers such that the following conditions hold
(i) numbers $a,b,c,a+b+c$ are pairwise coprime,
(ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power.
Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.
Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.
For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals:
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac27\qquad\textbf{(C)}\ \frac67\qquad\textbf{(D)}\ \frac{9}{32}\qquad\textbf{(E)}\ \frac{27}{32} $
In a country, there are \( n \) cities. Each pair of cities is connected either by a paved road or a dirt road. It is known that there exists a pair of cities such that it is impossible to travel between them using only paved roads. Show that, in this case, it is possible to travel between any two cities using only dirt roads.
If for function $f$ holds that $$f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0$$ for any $\in\mathbb{R}$, prove that $f$ is periodic and find one period of her.
Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya .
[img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]
A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.
Find the remainder when $2^{1990}$ is divided by $1990.$
A side of an arbitrary triangle has a length greater than $1$. Prove that the given triangle it can be cut into at least $2$ triangles, so that each of them has a side of length equal to $1$.
How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?
How many of
$ 11^2 \plus{} 13^2 \plus{} 17^2$, $ 24^2 \plus{} 25^2 \plus{} 26^2$, $ 12^2 \plus{} 24^2 \plus{} 36^2$, $ 11^2 \plus{} 12^2 \plus{} 132^2$ are perfect square ?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 d)1 \qquad\textbf{(E)}\ 0$
$1956$ points are chosen in a cube with edge $13$. Is it possible to fit inside the cube a cube with edge $1$ that would not contain any of the selected points?
A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices
are always on the $x$- and $y$-axes respectively. Find the locus of the midpoint of the square.
Let $ x_{1} ,x_{2} ,\ldots ,x_{9}$ be real numbers on $ \left[ \minus{} 1,1\right]$. If $ \sum _{i \equal{} 1}^{9}x_{i}^{3} \equal{} 0$, then what is the largest possible value of $ \sum _{i \equal{} 1}^{9}x_{i}$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac {3}{2} \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \frac {9}{2} \qquad\textbf{(E)}\ \text{None}$