International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria


Ivan is Watching You
Ivan's Office

Day 1
    Problem 1
    Problem 2
    Problem 3
    Problem 4
    Problem 5

Day 2
    Problem 6
    Problem 7
    Problem 8
    Problem 9
    Problem 10


    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions
    Closing Ceremony

Official IMC site

Problem 8

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.

(Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University)

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 \(\displaystyle \varphi(u,v)=\varphi(u-1,v)+\varphi(u,v-1)\).