Download [PDF] 2021 SMO Open Section Round 1 Questions Singapore Mathematical Olympiad

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Singapore Mathematical Society Singapore Mathematical Olympiad {SMO) 2021 (Open Section, Round 1) Thursday, 3 June 2021 0930-1200 hrs Instructions to contestants 1. Answer ALL 25 questions . 2. W叩 eyour answers in the answer sheet provided and shade the appropriate bubbles below your answers. 3. No steps are needed to justify your answers . 4. Each question carries 1 mark. 5 . No calculators are allowed . PL~ASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO 1

In this paper, let 股 denote the set of all real numbers, and l x」denote the greatest integer not exceeding x and let f x l denote the smallest integer not less than x. For examples, l 5」= 5, l2.8 」= 2, and l-2.3 」=-3; 「57 = 5, 「2.81 = 3, and 「-2.31 = -2. 7r 1. It is given that -2 u2021 sin(2a)I 」． < (3 < a < 31r 12 3 4 —, cos(a - /3) = - and sin(a + /3) = -- 13 5 Find 2. Find the number of solutions of the equation Ix - 31 + Ix - 51 = 2. (Note: If you think that there are infinitely many solutions, enter your answer 邸 "99999" .) 3. Evaluate 1 x 2 x 3 + 2 x 3 x 4 + 3 x 4 x 5 +· · ·+ 10 x 11 x 12. 4. It is given that the solution of the inequality 0 汀~:::; kx+ 1 is a:::; x:::; b with b-a = 2, where k > 0. Determine 忱」． 5. The figure below shows a cross that is cut out from a 10 x 9 rectangular board. Find the total number of rectangles in the above figure. (Note: A square is a rectangle.) 6. Consider all the polynomials P(x, y) in two variables such that P(O, 0) = 2020 and for all x and y, P(x, y) = P(x + y, y - x). Find the largest possible value of P(l, 1). 7. In the three dimensional Cartesian space with i, j and k denoting the unit vectors along three perpendicular directions in a clockwise manner, the line l with equation given by r x (i + 2j + 3k) = 5i -13j + 7k intersects the plane II with equation x + y + z = 16 at the point (a, b, c). Find the value of a+ b + c. 8. Find the minimum value of (x+ 7)红 (y+2)2 subject to the constraint (x-5) 红 (y-7)2 = 4. 2

9. Find the largest possible value of a红沪 +,4 among all possible sets of numbers (a,{3,-y) that satisfy the equations a+/3+-y = 2 a2+ 沪 +,2 = 14 a3+ 胪＋牡= 20. 10. If p is the product of all the non-zero real roots of the equation {/x1 + 30x5 = v'泸 -30x 气 find LIPI 」． 11. Let S be the sum of a convergent geometric series with first term 1. If the third term of the series is the arithmetic mean of the first two terms, find L 3S + 4」. 1 1 12 . Given that sin a + sin f3 =— and cos a+ cos/3 =-,find ltan 邵 +/3) 」． 10'9 13 . Determine the number of positive integers that are divisible by 2021 and has exactly 2021 divisors (including 1 and itself). 14. Lets= E (~0i) - 298. Find ll卢 IJ 3 15. Assume that ABC is an acute triangle with sin(A + B) = -and sin(A - B) = -. If 5 5 AB = 2022(/6 -2), determine L h」, where his the height of the triangle from Con AB. n+2 16. Let a1, a2, · · ·be a sequence with a1 = 1 and 0n+1 =一— -Sn for all n = l, 2, · · ·, where n Sn= a1 + a2 十．．．十 an. Determine the minimum integer n such that an 2:-: 2021. 17 . Each card of a stack of 101 cards has one side colored red and the other colored blue. Initially all cards have the red side facing up and stacked together in a deck. On each turn, Ah Meng takes 8 cards on the top, flip them over, and place them to the bottom deck. Determine the minimum number of turns required so that all the cards have the red sides facing up again . cosA AC 4 18 . Let ABC be a triangle with AB = 10 and ——- = - = -. Let P be a point on the cosB BC 3 inscribed circle of triangle ABC. Find the largest possible value of P A2 + P B2 + PC2. 19 . A basket contains 19 apples labeled by the numbers 2,3, .. .' ,20, and 19 bananas labeled by the numbers 2,3, .. . , 20. Ah Beng picks m apples and n bananas from the basket. However he needs to ensure that for any apple labeled a and any banana labeled b that he picks, a and b are relatively prime. Determine the largest possible value of mn . 3

20. Let p(x) = a夕- bx+ c be a polynomial where a, b, c are positive integers and p(x) has two distinct roots in (0, 1). Determine the least possible value of abc. 21 ln the triangle ABC, 乙A > 90°, the incircle touches the side BC and AC at A1 and B1 respectively . The line A1 趴 meets the extension of BA at X such that 乙CXB = 90° . Suppose BC2 = AB2 + BC·AC. Find the size of 乙A in degrees. 22. Find the number of positive integers n such that 7n - 16 divides n• 132019. 23. In the acute triangle ABC, Pis a point on AB, Q is a point on AC such that BP+CQ = PQ. The bisector of 乙A meets the circumcircle of the triangle ABC at the point R distinct from A. Suppose 乙PRQ = 52.5°. Find the size of 乙BAG in degrees. 00 24 . Let S = J 习工 ldx. Determine the value of LS 勺 -oo 25. Let p, q, r be positive numbers with p-r = 4q, and a1, a2, ···and b1, 切，· • • be two sequences defined by a1 = p, b1 = q and for n~2, an= Plln-1, bn = qan-1 + rb九一 1· Find the value of lim n➔oo ✓a~+ (3bn)2 b九 4