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1. Find the rightmost digit (that of the units) of 17^(17^(17)).
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2. Show that there is no surjective map between a set S and the set of its subsets (its power set) P(S).
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3. Show that 1996 has a multiple whose decimal representation contains the digit 4 only.
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4. Construct a number 0.a_1a_2... such that a_k=0 if k admits has an even number of prime factors, and a_k=1 otherwise. Is this number rational ?
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5. a,b,n are positive integers such that a <> b. Show that if n divides a^n-b^n, then n also divides (a^n-b^n)/(a-b).
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6. Find all functions f: Z -> Z such that f(x^3+y^3+z^3)=f(x)^3+f(y)^3+f(z)^3 for all integers x,y,z.
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7. We call Z-ball any subset of R^3 of the form E_R={(x,y,z) in Z^3; x^2+y^2+z^2 <= R^2}, where R is real. Show that a Z-ball cannot contain exactly 2005 points (implied: with integer coordinates).
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8. Let f: R -> R be a function of class C^2, such that |f| <= M and |f ''| <= M' for some M,M' in R. Show that |f '| <= sqrt(2MM').
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9. Two players play a game. Player 1 picks a binary digit b_0, then player 2 picks a binary digit b_1, etc ... Player 1 wins if and only if 0.b_0b_1... is transcendental. Who wins the game ?
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10. Find a single binary operation from which -,+,x,/ can all be derived.
Solution