Found problems: 33
On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.
A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.
Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.
We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.
There are ten congruent segments on a plane. Each intersection point divides every segment passing through it in the ratio $3:4$. Find the maximum number of intersection points.
A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.
(A. Zertsalov, D. Skrobot)
Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?
A positive integer $n$ is given. On the segment $[0,n]$ of the real line $m$ distinct segments whose endpoints have integer coordinates are chosen. It turned out that it is impossible to choose some of thos segments such that their total length is $n$ and their union is $[0,n]$
Find the maximum possible value of $m$