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Transcript
Putnam Practice: Number theory The first few problems here are exercises in using the pigenhole principle: if you have n+1 pigeons and n holes, you have to put at least two pigeons in the same hole. More generally, if you have mn + 1 pigeons and n holes, you have to put at least m + 1 pigeons in the same hole. (1) Let A be any set of 20 distinct integers chosen from the arithmetic progression 1, 4, 7, . . . , 100. Prove that there must be two distinct integers in A whose sum is 104. (2) A lattice point in the plane is a point (x, y) such that both x and y are integers. Find the smallest number n such that given n lattice points in the plane, there exist two whose midpoint is also a lattice point. (3) Show that some multiple of 2017 is a number all of whose digits are zeroes and ones. (4) Let α be an irrational number. Prove that there are infinitely many integer pairs (h, k) with k > 0 such that α − h < 1 . k k2 (5) Prove that, for every set X = {x1 , x2 , . . . , xn } of n real numbers, there exists a non-empty subset S of X and an integer m such that X 1 s ≤ . m + n+1 s∈S Next are three problems involving congruences and Fermat’s little theorem. Let n be a positive integer. The Euler function φ(n) is equal to the number of positive integers k = 1, 2, . . . , n−1 relatively prime to n. The Euler function can be easily calculated using that φ(pk ) = pk −pk−1 for p prime, and φ(mn) = φ(m)φ(n) when m and n are relatively prime. Fermat’s theorem states that if a is relatively prime to n, then aφ(n) ≡ 1 (mod n). There is also the Chinese remainder theorem: if m and n are two relatively prime integers, then the system of congruences x ≡ a (mod m) , x ≡ b (mod n) has a unique solution mod mn. (6) (7) 17 Deterrmine the last digit of 1717 . Find the last three digits of 79999 . 1 2 2 22 Show that for every n the sequence 2, 22 , 22 , 22 ,. . . (mod n) is eventually constant. Here are two more “themed” problems, involving Pell’s equation: if d is a positive non-square integer, then solutions of (8) x2 − dy 2 = 1 with positive x and y are (xk , yk ), k = 1, 2, . . . where (x1 , y1 ) is the solution with smallest x and y and √ √ xk + yk d = (x1 + y1 d)k . (9) (10) Show that there are infinitely many integers such that n2 + (n + 1)2 is a perfect square. Let m and n be two integers (1 ≤ m ≤ 1981, 1 ≤ n ≤ 1981) that satisfy the equation (n2 − nm − m2 )2 = 1. Find the maximal value for n2 + m2 . And here are three more (with no particular theme), the last one very hard. (11) A composite (positive integer) is a product ab with a and b not necessarily distinct integers in {2, 3, 4, . . .}. Show that every composite is expressible as xy + xz + yz + 1, with x, y, and z positive integers. (12) Let S be the set of all ordered triples (p, q, r) of prime numbers for which at least one rational number x satisfies px2 + qx + r = 0. Which primes appear in seven or more elements of S? (13) Let p be an odd prime number such that p ≡ 2 (mod 3). Define a permutation π of the residue classes modulo p by π(x) ≡ x3 (mod p). Show that π is an even permutation if and only if p ≡ 3 (mod 4).