International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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

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

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

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    Solution