Suppose that for real $x,y,z,t$ the following equalities hold:$\{x+y+z\}=\{y+z+t\}=\{z+t+x\}=\{t+x+y\}=1/4$. Find all possible values of $\{x+y+z+t\}$.(Here$\{x\}=x-[x]$)

Let $M$ be the midpoint of the side $BC$ of $\triangle ABC$. On the side $AB$ and $AC$ the points $E$ and $F$ are chosen. Let $K$ be the point of the intersection of $BF$ and $CE$ and $L$ be chosen in a way that $CL\parallel AB$ and $BL\parallel CE$. Let $N$ be the point of intersection of $AM$ and $CL$. Show that $KN$ is parallel to $FL$. Edit:Fixed typographical error.

It is known that for natural numbers $a,b,c,d$ and $n$ the following inequalities hold: $a+c<n$ and $a/b+c/d<1$. Prove that $a/b+c/d<1-1/n^3$.

There are $100$ cards with numbers from $1$ to $100$ on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number $n$ then Nick has a card with a number $2n+2$.What is the maximal number of cards that could be taken by the two guys?

Find the values of $x$ such that the following inequality holds: $\min\{\sin x,\cos x\}<\min\{1-\sin x,1-\cos x\}$

Find all pairs of prime numbers $p$ and $q$ that satisfy the equation $3p^{q}-2q^{p-1}=19$.

Is it possible to choose $24$ points in the space,such that no three of them lie on the same line and choose $2013$ planes in such a way that each plane passes through at least $3$ of the chosen points and each triple of points belongs to at least one of the chosen planes?

Let $M$ be the midpoint of the internal bisector $AD$ of $\triangle ABC$.Circle $\omega_1$ with diameter $AC$ intersects $BM$ at $E$ and circle $\omega_2$ with diameter $AB$ intersects $CM$ at $F$.Show that $B,E,F,C$ are concyclic.