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**apsoni123** Hello I am in Grade 10 and I am trying to solve the below question as mock paper for a math contest. Your help would be apperciated.

Consider the following two-player game. Starting with the number 0, players take turns adding to the current sum; on your turn, you can add either 4 or 7. If on your turn you can make the new sum end in two zeros (i.e., if your turn leaves a multiple of 100), you win.

Assuming best play, is there a winning strategy for either player, or will the game go on indefinitely? If there is a winning strategy, should you move first or second, and how do you play from there?

Suppose *r* and *s* are positive integers. Let *F* be a function from the set of all positive integers {1,2,3,…} to itself with the following properties:

*F* is one-to-one and onto. (If you don't know these terms, look them up online.)

For every positive integer *n*, either *F*(*n*) = *n* + *r* or *F*(*n*) = *n* − *s*.

If *r* = 5 and *s* = 8, what is *F*(2010)?

Find, with proof, the smallest positive integer *k* such that the *k*-fold composition of *F* with itself is the identity function; that is, *F*(*F*(…*F*(*n*))) = *n* for all *n* (where there are *k* copies of *F* on the left-hand side). The answer will depend on *r* and *s*.

For what values of *M* and *N* can an *M* × *N* chessboard be covered by an equal number of horizontal and vertical dominoes? (A domino always covers two adjacent squares on the board.)

If we tile the plane with black and white squares in a regular checkerboard pattern, then every square has an equal number (four) of black and white neighbors. (Two squares are considered neighbors if they are not the same but have at least one common point; squares that touch just at a vertex count as neighbors.) But if we try the analogous pattern of cubes in 3-dimensional space, it no longer works this way.

How many neighbors of each color does a white cube have?

Find a coloring pattern for a grid of cubes in 3-dimensional space so that every cube, whether black or white, has an equal number of black and white neighbors.

What happens in *n*-dimensional space for *n*>3? Is it still possible to find a color pattern for a grid of hypercubes, so that every hypercube, whether black or white, has an equal number of black and white neighbors?

I have about 4 more questions.

Thanks,

Abhi