Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
SW-ARML Practice 10-12-13 1. 2. 3. 4. 5. Find the value of a2 + a4 + a6 + + a100 if a1, a2, a3, …, a100 is an arithmetic sequence with common difference 1, and a1 + a2 + a3 + + a100 = 137. 200 What is the largest 2-digit prime factor of the integer ? 100 9 x 2 sin 2 x 4 Find the minimum value of for 0 < x < . x sin x A point P is chosen at random in the interior of a unit square S. Let d(P) denote the distance 1 1 m from P to the closest side of S. The probability that d ( P ) is equal to , where m 5 3 n and n are relatively prime positive integers. Find m + n. Positive numbers x, y, and z satisfy xyz = 1081 and log x (log yz) log y (log z) 168 . Find 6. log x log y 2 2 (log z)2 . Let N be the number of ordered pairs of nonempty sets A and B that have the following properties: A B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A B = , The number of elements of A is not an element of A, The number of elements of B is not an element of B. Find N. 7. Find the number of second-degree polynomials f(x) with integer coefficients and integer zeros for which f(0) = 2013. 8. Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is k% acid. From jar C, m liters of the solution is added to jar A, and the remainder of the solution in jar C is added n to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that m and n are relatively prime positive integers, find k + m + n. 9. What is the product of the real roots of the equation x2 + 18x + 30 = 2 x2 18x 45 ? 10. Let an = 6n + 8n for all positive integers n. Determine the remainder upon dividing a83 by 49. 1