IMONST = International Mathematical Olympiad National Selection Test
Malaysia 2020 Round 1 JuniorsTime: 2.5 hours
∙ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
∙ No mark is deducted for a wrong answer.
∙ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.
Part A (1 point each)
p1. The number N is the smallest positive integer with the sum of its digits equal to 2020. What is the first (leftmost) digit of N?
p2. At a food stall, the price of 16 banana fritters is k RM , and the price of k banana fritters is 1 RM . What is the price of one banana fritter, in sen?Note: 1 RM is equal to 100 sen.
p3. Given a trapezium ABCD with AD∥ to BC, and ∠A=∠B=90o. It is known that the area of the trapezium is 3 times the area of △ABD. Find areaof△ABDareaof△ABC.
p4. Each △ symbol in the expression below can be substituted either with + or −: △1△2△3△4. How many possible values are there for the resulting arithmetic expression?Note: One possible value is −2, which equals −1−2−3+4.
p5. How many 3-digit numbers have its sum of digits equal to 4?
Part B (2 points each)
p6. Find the value of +1+2+3−4−5−6+7+8+9−10−11−12+...−2020 where the sign alternates between + and − after every three numbers.
p7. If Natalie cuts a round pizza with 4 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.
p8. Given a square with area A. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area B lies inside the circle, such that all its vertices lie on the circle. Find the value of A/B.
p9. This sequence lists the perfect squares in increasing order: 0,1,4,9,16,...,a,108,b,... Determine the value of b−a.
p10. Determine the last digit of 55+66+77+88+99.
Part C (3 points each)
p11. Find the sum of all integers between −1442 and 2020.
p12. 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 3 months if I work alone". Wahib says, "I can paint this building in 2 months if I work alone". Wahub says, "I can paint this building in k months if I work alone". If they work together, they can finish painting the building in 1 month only. What is k?
p13. Given a rectangle ABCD with a point P inside it. It is known that PA=17, PB=15, and PC=6. What is the length of PD?
p14. What is the smallest positive multiple of 225 that can be written using digits 0 and 1 only?
p15. Given positive integers a,b, and c with a+b+c=20. Determine the number of possible integer values for ca+b.
Part D (4 points each)
p16. If we divide 2020 by a prime p, the remainder is 6. Determine the largest possible value of p.
p17. A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by 5 hexagons. Each hexagons is bordered by 3 pentagons and 3 hexagons. We need 12 pentagons to make one football. How many hexagons are needed to make one football?
p18. Given a right-angled triangle with perimeter 18. The sum of the squares of the three side lengths is 128. What is the area of the triangle?
p19. A perfect square ends with the same two digits. How many possible values of this digit are there?
p20. Find the sum of all integers n that fulfill the equation 2n(6−n)=8n. algebrageometrycombinatoricsnumber theory