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Part of 2021 Malaysia IMONST 1

Problems(1)

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

Source:

9/21/2021
IMONST = International Mathematical Olympiad National Selection Test Malaysia 2021 Round 1 Juniors
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. 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?
p2. 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?
p3. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of 1414 wings, 2424 heads, and 6262 legs. How many goldshes are there inside Clarissa's shop?
p4. 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?
p5. 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 B (2 points each)
p6. 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?
p7. 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.
p8. Haz 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?
p9. What is the smallest positive multiple of 2424 that can be written using digits 44 and 55 only?
p10. 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?
Part C (3 points each)
p11. Dinesh has several squares and regular pentagons, all with side length 1 1. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg
p12. If x+1x=5x +\frac{1}{x} = 5, what is the value of x3+1x3x^3 +\frac{1}{x^3} ?
p13. There are 1010 girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?
p14. The two diagonals of a rhombus have lengths with ratio 3:43 : 4 and sum 5656. What is the perimeter of the rhombus?
p15. How many integers nn (with 1≤n≤20211 \le n \le 2021) have the property that 8n+18n + 1 is a perfect square?
Part D (4 points each)
p16. Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is 2424. The length between the midpoint of the straight edge and the midpoint of the arc is 66. Find the radius of the circle.
p17. Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is 1818. How many passcodes satisfy these conditions?
p18. A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day?
p19. Find the sum of (decimal) digits of the number (102021+2021)2(10^{2021} + 2021)^2?
p20. Determine the number of integer solutions (x,y,z)(x, y, z), with 0≤x,y,z≤1000 \le x, y, z \le 100, for the equation(x−y)2+(y+z)2=(x+y)2+(y−z)2.(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.
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