Hanoi Open Mathematical Competition 2014 Senior Section Sunday, 23 March 2014 08h30-11h30 Important: Answer all 15 questions Enter your answers on the answer sheet provided No calculators are allowed Q1 Let a and b satisfy the conditions a3 − 6a2 + 15a = b3 − 3b2 + 6b = −1 The value of (a − b)2014 is (A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above Q2 How many integers are there in the set {0, 1, 2, , 2014} such that x 999 C2014 ≥ C2014 (A): 15; (B): 16; (C): 17; (D): 18; (E) None of the above Q3 How many 0’s are there in the sequence x1 , x2 , x3 , , x2014 , where n+1 n xn = √ − √ , n = 1, 2, , 2014 2015 2015 (A): 20; (B): 30; (C): 40; (D): 50; (E) None of the above Q4 Find the smallest positive integer n such that the number 2n + 28 + 211 is a perfect square (A): 8; (B): 9; (C): 10; (D): 11; (E) None of the above Q5 The first two terms of a sequence are and Each next term thereafter is the sum of the nearest previous two terms if their sum is not greater than 10 and is otherwise The 2014th term is (A): 0; (B): 8; (C): 6; (D): 4; (E) None of the above Q6 Let S be area of the given parallelogram ABCD and the points E, F belong to BC and AD, respectively, such that BC = 3BE, 3AD = 4AF Let O be the intersection of AE and BF Each straightline of AE and BF meets that of CD at points M and N, respectively Determine area of triangle M ON Q7 Let two circles C1 , C2 with different radius be externally tangent at a point T Let A be on C1 and B be on C2 , with A, B = T such that ∠AT B = 900 (a) Prove that all such lines AB are concurrent (b) Find the locus of the midpoints of all such segments AB Q8 Determine the integral part of A, where A= 1 + + ··· + 672 673 2014 Q9 Solve the system 16x3 + 4x = 16y + 16y + 4y = 16x + Q10 Find all pairs of integers (x, y) satisfying the condition 12x2 + 6xy + 3y = 28(x + y) Q11 Determine all real numbers a, b, c, d such that the polynomial f (x) = ax3 + bx2 + cx + d satisfies simultaneously the folloving conditions |f (x)| ≤ for |x| ≤ f (2) = 26 Q12 Given rectangle paper of size 15 cm × 20 cm, fold it along a diagonal Determine the area of the common part of two halfs of the paper? Q13 Let a, b, c satisfy the conditions 5 ≥ a ≥ b ≥ c ≥ a+b≤8 a + b + c = 10 Prove that a2 + b2 + c2 ≤ 38 Q14 Let ω be a circle with centre O, and let be a line that does not intersect ω Let P be an arbitrary point on Let A, B denote the tangent points of the tangent lines from P Prove that AB passes through a point being independent of choosing P Q15 Let a1 , a2 , , a9 ≥ −1 and a31 + a32 + · · · + a39 = Determine the greatest value of M = a1 + a2 + · · · + a9 —————————————–