By Serguei A. Stepanov

ISBN-10: 0306110369

ISBN-13: 9780306110368

Writer S.A. Stepanov completely investigates the present nation of the idea of Diophantine equations and its similar equipment. Discussions concentrate on mathematics, algebraic-geometric, and logical elements of the challenge. Designed for college students in addition to researchers, the booklet contains over 250 excercises followed by way of tricks, directions, and references. Written in a transparent demeanour, this article doesn't require readers to have distinctive wisdom of contemporary equipment of algebraic geometry.

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**Extra info for Arithmetic of algebraic curves**

**Example text**

If the sequence {a,} is defined by then a100 equals (A) 9900 (B) 9902 (C) 9904 (D) 10100 (E) 10102 14. The product of all real roots of the equation x'Og1oZ = 10 is (A) 1 (B) -1 (C) 10 (D) lo-' (E) none of these 15. If sin22 sin32 = cos2x cos3x, then one value for x is (A) 18" (B) 30" (C) 36" (D) 45" (E) 60" 16. The function f(x) satisfies f(2+x) = f(2-x) for all real numbers x. If the equation f(x) = 0 has exactly four distinct real roots, then the sum of these roots is CONTEST PROBLEM BOOK V 10 A 17.

Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative. (A) Dick, Ann, Carol, Bill (C) Dick, Carol, Bill, Ann (E) Ann, Dick, Bill, Carol (B) Dick, Ann, Bill, Carol (D) Ann, Dick, Carol, Bill 18. It takes A algebra books (all the same thickness) and H geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, S of the algebra books and M of the geometry books would fill the same shelf.

What is the probability that, among those selected, the second smallest is 3? (A) 1/60 (B) 1/6 (C) 1/3 (D) 1/2 (E) none of these CONTESTPROBLEM BOOK V 24 23. Let + N = 6g5 5 * 6g4 + 10 - 6g3 + 10 * 69' +5 How many positive integers are factors of N? (A) 3 (B)5 (C) 69 * 69 + 1. (E) 216 (D) 125 24. Let p(x) = x2 + bx + c, where b and c are integers. If p(x) is a factor of both x4 + 62' + 25 + and 3x4 4x2 + 282 + 5, what is p ( 1) ? (A) 0 (B) 1 (C) 2 (D)4 (El 8 25. )J 26. It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the x and y axes and so that the medians to the midpoints of the legs lie on the lines y = 32 + 1 and y = mx+2.

### Arithmetic of algebraic curves by Serguei A. Stepanov

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