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2020 Malaysia IMONST 1 Primary 20 problems 2.5 hours, integer >=0 answer only

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September 20, 2021
algebrageometrynumber theorycombinatorics

Problem Statement

International Mathematical Olympiad National Selection Test Malaysia 2020 Round 1 Primary Time: 2.5 hours ∙\bullet For each problem you have to submit the answer only. The answer to each problem is a non-negative integer. ∙\bullet No mark is deducted for a wrong answer. ∙\bullet The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.
Part A (1 point each)
p1. Annie asks his brother four questions, "What is 2020 plus 2020? What is 2020 minus 2020? What is 2020 times 2020? What is 2020 divided by 2020?". His brother adds the answers to these four questions, and then takes the (positive) square root of the result. What number does he get?
p2. A broken watch moves slower than a regular watch. In every 77 hours, the broken watch lags behind a regular watch by 1010 minutes. In one week, how many hours does the broken watch lags behind a regular watch?
p3. Given a square ABCDABCD. A point PP is chosen outside the square so that triangle BCPBCP is equilateral. Find ∠APC\angle APC, in degrees.
p4. Hussein throws 4 dice simultaneously, and then adds the number of dots facing up on all 44 dice. How many possible sums can Hussein get?
Note: For example, he can get sum 1414, by throwing 44, 66, 33, and 1 1. Assume these are regular dice, with 11 to 66 dots on the faces.
p5. Mrs. Sheila says, "I have 55 children. They were born one by one every 33 years. The age of my oldest child is 77 times the age of my youngest child." What is the age of her third child?
Part B (2 points each)
p6. The number NN is the smallest positive integer with the sum of its digits equal to 20202020. What is the first (leftmost) digit of NN?
p7. At a food stall, the price of 1616 banana fritters is kk RM , and the price of kk banana fritters is 1 1 RM . What is the price of one banana fritter, in sen?
Note: 11 RM is equal to 100100 sen.
p8. Given a trapezium ABCDABCD with AD∥AD \parallel to BCBC, and ∠A=∠B=90o\angle A = \angle B = 90^o. It is known that the area of the trapezium is 3 times the area of △ABD\vartriangle ABD. Findarea  of  △ABCarea  of  △ABD.\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.
p9. Each △\vartriangle symbol in the expression below can be substituted either with ++ or −-:△1△2△3△4.\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.How many possible values are there for the resulting arithmetic expression?
Note: One possible value is −2-2, which equals −1−2−3+4-1 - 2 - 3 + 4.
p10. How many 33-digit numbers have its sum of digits equal to 44?
Part C (3 points each)
p11. Find the value of+1+2+3−4−5−6+7+8+9−10−11−12+...−2020+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020where the sign alternates between ++ and −- after every three numbers.
p12. If Natalie cuts a round pizza with 44 straight cuts, what is the maximum number of pieces that she can get?
Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.
p13. Given a square with area A A. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area B B lies inside the circle, such that all its vertices lie on the circle. Find the value of A/BA/B.
p14. This sequence lists the perfect squares in increasing order:0,1,4,9,16,...,a,108,b,...0, 1, 4, 9, 16, ... ,a, 10^8, b, ...Determine the value of b−ab - a.
p15. Determine the last digit of 55+66+77+88+995^5 + 6^6 + 7^7 + 8^8 + 9^9
Part D (4 points each)
p16. Find the sum of all integers between −1442-\sqrt{1442} and 2020\sqrt{2020}.
p17. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in 33 months if I work alone". Wahib says, "I can paint this building in 22 months if I work alone". Wahub says, "I can paint this building in kk months if I work alone". If they work together, they can finish painting the building in 11 month only. What is kk?
p18. Given a rectangle ABCDABCD with a point PP inside it. It is known that PA=17PA = 17, PB=15PB = 15, and PC=6PC = 6. What is the length of PDPD?
p19. What is the smallest positive multiple of 225225 that can be written using digits 00 and 1 1 only?
p20. Given positive integers a,ba, b, and cc with a+b+c=20a + b + c = 20. Determine the number of possible integer values for a+bc\frac{a + b}{c}.
PS. Problems 6-20 were also used in [url=https://artofproblemsolving.com/community/c4h2675966p23194287]Juniors as 1-15. Problems 11-20 were also used in Seniors 1-10.
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