Find minimum number $n$ that: 1) $80|n$ 2) we can permute 2 different numbers in $n$ to get $m$ and $80|m$

$\omega$ is incircle of $\triangle ABC$ touch $AC$ in $S$. Point $Q$ lies on $\omega$ and midpoints of $AQ$ and $QC$ lies on $\omega$ . Prove that $QS$ bisects $\angle AQC$

Let $x_0$ - is positive root of $x^{2017}-x-1=0$ and $y_0$ - is positive root of $y^{4034}-y=3x_0$ a) Compare $x_0$ and $y_0$ b) Find tenth digit after decimal mark in decimal representation of $|x_0-y_0|$

3 cyclists rides on track in form circle with length $300$ meters in one direction. Every has constant speed,and speeds are different. Photographer want to make photoshoot with 3 cyclists. It is possible if they will be on the part of track with length $d$ meters. Find minimum $d$ such that it is possible.

$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?

There are $36$ gangsters bands.And there are war between some bands. Every gangster can belongs to several bands and every 2 gangsters belongs to different set of bands. Gangster can not be in feuding bands. Also for every gangster is true, that every band, where this gangster is not in, is in war with some band, where this gangster is in. What is maximum number of gangsters in city?

$(a_i),(b_i)$ are nonconstant arithmetic and geometric progressions. $a_1=b_1,a_2/b_2=2,a_4/b_4=8$ Find $a_3/b_3$.

Are there such $x,y$ that $\lg{(x+y)}=\lg x \lg y$ and $\lg{(x-y)}=\frac{\lg x}{\lg y}$ ?

There are $80$ peoples, one of them is murderer, and other one is witness of crime. Every day detective interrogates some peoples from this group. Witness will says about crime only if murderer will not be in interrogatory with him. It is enough $12$ days to find murderer ?

Point $D$ lies in $\triangle ABC$ and $BD=CD$,$\angle BDC=120$. Point $E$ lies outside $ABC$ and $AE=CE,\angle AEC=60$. Points $B$ and $E$ lies on different sides of $AC$. $F$ is midpoint $BE$. Prove, that $\angle AFD=90$

There is one nonzero digit in every cell of $2017\times 2017 $ table. On the board we writes $4034$ numbers that are rows and columns of table. It is known, that $4033$ numbers are divisible by prime $p$ and last is not divisible by $p$. Find all possible values of $p$. ExampleExample for $2\times2$. If table is |1|4| |3|7|. Then numbers on board are $14,37,13,47$