Combinatorics · Board processes · Monovariants · Invariants · Harmonic mean · Powers of two
The number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until a single number is left on the board. Prove that this last number is not less than \(\frac{1}{2005}\).
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Serbian Republic Competition (Republicko takmicenje) 2005, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source