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2017-04-03 Advanced


Does there exist a continuous function f:R2Rf : \mathbb{R}^2 \to \mathbb{R} that is also injective?

An injective function is one-to-one, meaning that no element of the range is the image of more than one element of the domain.

8 resistors (orange color) are connected to form a regular octagon. 8 more resistors (blue color) connect the vertices of the octagon to its center. All the 16 resistors are of resistance 420 Ω\SI{420}{\ohm}.

If the connecting wires have negligible resistance, calculate the equivalent resistance (in ohms, rounded to the nearest integer) between the terminals AA and BB.

Find the smallest positive integer nn for which

n50+(n+1)50>(n+2)50.n^{50} + (n+1)^{50} \gt (n+2)^{50}.




Enter your answer as a.b a . b , where aa is the number of disconnected regions, and bb is the number of regions which have infinite volume. For example, if you believe there are 4 regions, of which 3 have infinite volume, enter your answer as 4.34.3.


On a large 999×999 999 \times 999 board, each cell is colored either white or black.

Define a coordinated triple of cells (C1,C2,C3) (C_1, C_2, C_3 ) as the formation such that

  • C1C_1 and C2C_2 are in the same row;
  • C2C_2 and C3C_3 are in the same column;
  • C1C_1 and C3C_3 are white;
  • C2 C_2 is black.

What is the maximum number of coordinated triples? Enter your answer as the last three digits of that maximum.


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