Find the sum of all positive integers $n$ such that $$\frac{n+11}{\sqrt{n-1}}$$is an integer.

The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$. What are the possible numbers u can get after $99$ consecutive operations of these?

Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find $$\frac{[ABKM]}{[ABCL]}$$

a) Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance? b) Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.

Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$

Inside a rhombus $ABCD$ with $\angle BAD=60$, points $F,H,G$ are choosen on lines $AD,DC,AC$ respectivily such that $DFGH$ is a paralelogram. Show that $BFH$ is a equilateral triangle.

Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules: $\cdot$ The frog can jump only in points of $M$ $\cdot$ The frog can't jump more than $1$ time over the same point. $\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$ In how many ways the Frog can make his target?