MathDB
Indonesia Regional MO 2003 Part A 20 problems 90' , answer only

Source:

October 1, 2021
algebracombinatoricsnumber theorygeometryIndonesia Regional MO

Problem Statement

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2003 [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c2476068_2003_indonesia_regional]here
Time: 90 minutes ∙\bullet For each problem you have to submit the answer only. ∙\bullet Each correct answer is given a value of 1 and the question that is left blank without an answer or an incorrect answer is given a value of 0.

p1. If aa and bb are odd integers with a>ba > b, how many even integers are there between aa and bb ?
p2. Agung found that the average score of the three math tests he followed was 8181. The first test score was 8585. The third test score was 44 lower than the second test score. What is the value of Agung's second test?
p3. What is the set of solutions to the equation ∣x+2∣+∣3x∣=14|x + 2| + |3x| = 14 ?
p4. The four numbers 3,5,73, 5, 7 and 88 will be entered into the boxes on the side. What is the greatest yield that can be obtained? https://cdn.artofproblemsolving.com/attachments/a/b/2152b5a3dcc4634087fcc1d535131cae0f30a4.png
p5. Let x,y,zx, y, z be three different natural numbers. The third greatest common divisor is 12 12, while the third least common multiple is 840840. What is the greatest value for x+y+zx + y + z ?
p6. What is the smallest positive integer kk such that 20032003...2003⏟k\underbrace{20032003...2003}_{k} is divisible by 99 ?
p7. The quadratic equation 2x2−2(2a+1)x+a(a−1)=02x^2-2(2a + 1)x + a(a-1) = 0 has two real roots x1x_1 and x2x_2. What value of a satisfies the quadratic equation so that x1<a<x2x_1 < a < x_2 ?
p8. In an isosceles triangle ABCABC, a square PQRSPQRS is made as follows: Point PP is on side ABAB, point QQ is on side ACAC, while points RR and SS are on hypotenuse BCBC. If the area of ​​triangle ABCABC is xx, what is the area of ​​the square PQRSPQRS ?
p9. Upik throws nn dice. He calculates the probability that the sum of the dice is 66. For nn what is the greatest probability?
p10. A vertical line divides a triangle with vertices (0,0)(0,0), (1,1)(1,1) and (9,1)(9,1) into two areas of equal area. What is the equation of the line?
p11. Let mm and nn be two natural numbers that satisfy m2−2003=n2m^2-2003 = n^2. Calculate mnmn.
p12. What is the value of x that satisfies 4log⁡(2log⁡x)+2log⁡(4log⁡x)=2^4 \log (^2 \log x) + ^2 \log (^4 \log x) = 2 ?
p13. Point PP lies inside the square ABCDABCD such that AP:BP:CP=1:2:3AP : BP : CP = 1: 2: 3. What is the measure of angle APBAPB ?
p14. By combining the three basic colors red, yellow, and blue other colors can be formed. Suppose there are 55 cans of red paint, 55 cans of yellow, and 55 cans of blue. Budi can choose any can to mix colors, and all paint in a can must be used all. How many color choices are there?
p15. Mr. Oto bought two cars for resale. He made a 30%30\% profit on the first car, but suffered a 20%20\% loss on the second car. The selling price for both cars is the same. What is the percentage profit (or loss) of Pak Oto as a whole? [Note: All percentage to the purchase price. For the answer, use the '-' sign to represent the loss and the '+' sign to represent the gain.]
p16. Four married couples watch an orchestra performance. Their seats must be separated between the husband's group and the wife's group. For each group, there are 4 seats next to each other in a row. How many ways are there to give them a seat?
p17. A ball with fingers rr is kicked from BB to AA. The ball rolls exactly 1010 laps before hitting an inclined plane and stopping. What is the distance from BB to AA? https://cdn.artofproblemsolving.com/attachments/e/5/f5e9b117e0815601fdf0b0b046607d798e10fd.png
p18. What is the remainder of the division 1⋅1!+2⋅2!+3⋅!+...+99⋅99!+100⋅100!1\cdot 1! + 2\cdot 2! + 3\cdot ! + ... + 99\cdot 99! + 100\cdot 100! by 101101?
p19. A circle has a diameter ABAB whose length is a 22-digit integer. The arc string CDCD is perpendicular to ABAB and intersects ABAB at point HH. The length of CDCD is equal to the number obtained by changing the position of the two digits from the length of ABAB. If the distance from HH to the center of the circle is a rational number, what is the length of ABAB?
p20. How many ways to choose three different numbers so that there are no two consecutive numbers, if the numbers are chosen from the set {1,2,3,...,9,10}\{1, 2, 3,..., 9, 10\}?
Back to ProblemsView on AoPS