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Part of 2021 Malaysia IMONST 1

Problems(1)

2021 Malaysia IMONST 1 Primary 20 problems 2.5 hours, integer >=0 answer only

Source:

9/21/2021
International Mathematical Olympiad National Selection Test Malaysia 2021 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. Faris has six cubes on his table. The cubes have a total volume of 20212021 cm3^3. Five of the cubes have side lengths 55 cm, 55 cm, 66 cm, 66 cm, and 1111 cm. What is the side length of the sixth cube (in cm)?
p2. What is the sum of the first 200200 even positive integers?
p3. Anushri writes down five positive integers on a paper. The numbers are all different, and are all smaller than 1010. If we add any two of the numbers on the paper, then the result is never 1010. What is the number that Anushri writes down for certain?
p4. If the time now is 10.0010.00 AM, what is the time 1,0001,000 hours from now? Note: Enter the answer in a 1212-hour system, without minutes and AM/PM. For example, if the answer is 9.009.00 PM, just enter 99.
p5. Aminah owns a car worth 10,00010,000 RM. She sells it to Neesha at a 10%10\% profit. Neesha sells the car back to Aminah at a 10%10\% loss. How much money did Aminah make from the two transactions, in RM?
Part B (2 points each)
p6. Alvin takes 250 small cubes of side length 11 cm and glues them together to make a cuboid of size 55 cm  ×5\times 5 cm  ×10\times 10 cm. He paints all the faces of the large cuboid with the color green. How many of the small cubes are painted by Alvin?
p7. Cikgu Emma and Cikgu Tan select one integer each (the integers do not have to be positive). The product of the two integers they selected is 20212021. How many possible integers could have been selected by Cikgu Emma?
p8. A three-digit number is called superb if the first digit is equal to the sum of the other two digits. For example, 431431 and 909909 are superb numbers. How many superb numbers are there?
p9. Given positive integers a,b,ca, b, c, and dd that satisfy the equation 4a=5b=6c=7d4a = 5b =6c = 7d. What is the smallest possible value of b b?
p10. Find the smallest positive integer n such that the digit sum of n is divisible by 55, and the digit sum of n+1n + 1 is also divisible by 55.
Note: The digit sum of 14401440 is 1+4+4+0=91 + 4 + 4 + 0 = 9.
Part C (3 points each)
p11. Adam draws 77 circles on a paper, with radii 1 1 cm, 22 cm, 33 cm, 44 cm, 55 cm, 66 cm, and 77 cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm2^2) between the total area of the red circles and the total area of the blue circles?
p12. The number 20212021 has a special property that the sum of any two neighboring digits in the number is a prime number (2+0=22 + 0 = 2, 0+2=20 + 2 = 2, and 2+1=32 + 1 = 3 are all prime numbers). Among numbers from 20212021 to 20412041, how many of them have this property?
p13. Clarissa opens a pet shop that sells three types of pets: gold shes, hamsters, and parrots. The pets inside the shop together have a total of 1414 wings, 2424 heads, and 6262 legs. How many gold shes are there inside Clarissa's shop?
p14. A positive integer nn is called special if nn is divisible by 44, n+1n+1 is divisible by 55, and n+2n + 2 is divisible by 66. How many special integers smaller than 10001000 are there?
p15. Suppose that this decade begins on 1 1 January 20202020 (which is a Wednesday) and the next decade begins on 1 1 January 20302030. How many Wednesdays are there in this decade?
Part D (4 points each)
p16. Given an isosceles triangle ABCABC with AB=ACAB = AC. Let D be a point on ABAB such that CDCD is the bisector of ∠ACB\angle ACB. If CB=CDCB = CD, what is ∠ADC\angle ADC, in degrees?
p17. Determine the number of isosceles triangles with the following properties: all the sides have integer lengths (in cm), and the longest side has length 2121 cm.
p18. Ha z marks kk points on the circumference of a circle. He connects every point to every other point with straight lines. If there are 210210 lines formed, what is kk?
p19. What is the smallest positive multiple of 2424 that can be written using digits 44 and 55 only?
p20. In a mathematical competition, there are 20212021 participants. Gold, silver, and bronze medals are awarded to the winners as follows: (i) the number of silver medals is at least twice the number of gold medals, (ii) the number of bronze medals is at least twice the number of silver medals, (iii) the number of all medals is not more than 40%40\% of the number of participants. The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded?
PS. Problems 11-20 were also used in [url=https://artofproblemsolving.com/community/c4h2676837p23203256]Juniors as 1-10.
algebrageometrycombinatoricsnumber theory