104 number theory problems. From the training of the USA IMO by Titu Andreescu

By Titu Andreescu

The booklet is dedicated to the houses of conics (plane curves of moment measure) that may be formulated and proved utilizing in simple terms easy geometry. beginning with the well known optical homes of conics, the authors flow to much less trivial effects, either classical and modern. particularly, the bankruptcy on projective homes of conics incorporates a specified research of the polar correspondence, pencils of conics, and the Poncelet theorem. within the bankruptcy on metric houses of conics the authors talk about, specifically, inscribed conics, normals to conics, and the Poncelet theorem for confocal ellipses. The booklet demonstrates the good thing about in basic terms geometric equipment of learning conics. It includes over 50 routines and difficulties aimed toward advancing geometric instinct of the reader. The publication additionally comprises greater than a hundred conscientiously ready figures, as a way to aid the reader to higher comprehend the fabric provided

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Extra resources for 104 number theory problems. From the training of the USA IMO team

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We first look for d such that a 3 ≡ 1 (mod d) for all integers a. Fermat’s little theorem states that a p−1 ≡ 1 (mod p) for prime p and integers a relatively prime to p. If we set p − 1 = 3, we have p = 4, which is not a prime. Hence we cannot apply Fermat’s little theorem directly. On the other hand, if we set p = 7, then a 6 ≡ 1 (mod 7) for integers relatively prime to 7. It not difficult to check that the possible residue classes for a 3 modulo 7 are 0, 1, −1 (or 6). Hence, modulo 7, the possible residue classes for a 3 + b3 are 0, 1, −1, 2, −2.

Proof: Since x y = 10x + y and yx = 10y + x, their sum is equal to 11x +11y = 11(x + y), a composite number. 41. [AHSME 1973] In the following equation, each of the letters represents uniquely a different digit in base ten: (Y E) · (M E) = T T T. Determine the sum E + M + T + Y . Solution: Because T T T = T ·111 = T ·3·37, one of Y E and M E is 37, implying that E = 7. But T is a digit and T ·3 is a two-digit number ending with 7, and so it follows that T = 9 and T T T = 999 = 27·37, and so E +M+T +Y = 2+3+7+9 = 21.

We are basically considering all integers in the residue class a modulo d. We want to limit the number of residue classes that are cubes modulo d. In this way, we limit the number of residue classes that can be written as the sum of cubes modulo d. We first look for d such that a 3 ≡ 1 (mod d) for all integers a. Fermat’s little theorem states that a p−1 ≡ 1 (mod p) for prime p and integers a relatively prime to p. If we set p − 1 = 3, we have p = 4, which is not a prime. Hence we cannot apply Fermat’s little theorem directly.

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