International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

Home

Results
Individuals
Teams
Problem 8. Let $\displaystyle \Omega=\{(x,y,z)\in \mathbb{Z}^3: y+1\ge x\ge y\ge z\ge 0\}$. A frog moves along the points of $\displaystyle \Omega$ by jumps of length $\displaystyle 1$. For every positive integer $\displaystyle n$, determine the number of paths the frog can take to reach $\displaystyle (n,n,n)$ starting from $\displaystyle (0,0,0)$ in exactly $\displaystyle 3n$ jumps.
Hint: Project $\displaystyle \Omega$ to the plane by the map $\displaystyle (x,y,z)\mapsto(x+y,z)$ and consider the functions $\displaystyle \varphi$ on the lattice that satisfy $\displaystyle \varphi(u,v)=\varphi(u-1,v)+\varphi(u,v-1)$.