Rmo Test 1.docx

9/21/2017 10:13:00 PM Solve the equation y3 = x3 + 8x2 − 6x + 8, for positive integers x and y. Suppose ⟨ x1,x2,...,xn,...⟩ is a sequence of positive real numbers such that x1 ≥ x2 ≥ x3 ≥···≥xn···,andforalln x1 +x4 +x9 +···+xn2 ≤1. 123 n Show that for all k the following inequality is satisfied: x1 +x2 +x3 +···+xk ≤3. All the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by 5, are arranged in the increasing order. Find the 2000-th number in this list. Find all real values of a for which the equation x4 −2ax2 +x+a2 −a = 0 has all its roots real. Find all primes p and q such that p2 + 7pq + q2 is the square of an integer. Find the number of positive integers x which satisfy the condition If x, y, z are the sides of a triangle, then prove that |x2(y − z) + y2(z − x) + z2(x − y)| < xyz. Prove that the product of the first 200 positive even integers differs from the product of the first 200 positive odd integers by a multiple of 401. Solve the following equation for real x: (x2 +x−2)3 +(2x2 −x−1)3 =27(x2 −1)3. Let a, b, c be positive integers such that a divides b2, b divides c2 and c divides a2. Prove that abc divides (a + b + c)7. Suppose the integers 1,2,3,...,10 are split into two disjoint collections a1,a2,a3,a4,a5 and b1,b2,b3,b4,b5 suchthat a1 b2 >b3 >b4 >b5.

(i) Show that the larger number in any pair {aj,bj}, 1 ≤ j ≤ 5, is at least 6. (ii) Show that |a1 −b1|+|a2 −b2|+|a3 −b3|+|a4 −b4|+|a5 −b5| = 25 for every such partition. For any natural number n > 1, prove the inequality: 1<1+2+3+···+n<1+1. 2 n2 + 1 n2 + 2 n2 + 3 n2 + n 2 2n Find all integers a, b, c, d satisfying the following relations: (i) 1≤a≤b≤c≤d; (ii) ab+cd=a+b+c+d+3. If n is an integer greater than 7, prove that n − n is divisible by 7. Here n 777 denotes the number of ways of choosing 7 objects from among n objects; also, for any real number x, [x] denotes the greatest integer not exceeding x. Let a, b, c be three positive real numbers such that a + b + c = 1. Prove that among the three numbers a−ab, b−bc, c−ca there is one which is at most 1/4 and there is one which is at least 2/9. Find the number of ordered triples (x,y,z) of nonnegative integers satisfying the conditions: (i) x≤y≤z; (ii) x+y+z≤100. Find all real numbers a for which the equation x2 +(a−2)x+1=3|x| has exactly three distinct real solutions in x. CRMO 2004 link: https://homeworksonline.files.wordpress.com/2016/02/regionalmathematical-olympiad-2004-sol.pdf CRMO 2005 link:

https://homeworksonline.files.wordpress.com/2016/02/regionalmathematical-olympiad-2005 -sol.pdf \

9/21/2017 10:13:00 PM

9/21/2017 10:13:00 PM