MathDB

day 1

Part of 2003 Indonesia Juniors

Problems(1)

Indonesia Juniors 2003 day 1 OSN SMP

Source:

10/30/2021
p1. The pattern ABCCCDDDDABBCCCDDDDABBCCCDDDD...ABCCCDDDDABBCCCDDDDABBCCCDDDD... repeats to infinity. Which letter ranks in place 25332533 ?
p2. Prove that if a>2a > 2 and b>3b > 3 then ab+6>3a+2bab + 6 > 3a + 2b.
p3. Given a rectangle ABCDABCD with size 1616 cm ×25\times 25 cm, EBFGEBFG is kite, and the length of AE=5AE = 5 cm. Determine the length of EFEF. https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.png
p4. Consider the following series of statements. It is known that x=1x = 1. Since x=1x = 1 then x2=1x^2 = 1. So x2=xx^2 = x. As a result, x2−1=x−1x^2 - 1 = x- 1 (x−1)(x+1)=(x−1)⋅1(x -1) (x + 1) = (x - 1) \cdot 1 Using the rule out, we get x+1=1x + 1 = 1 1+1=11 + 1 = 1 2=12 = 1 The question. a. If 2=12 = 1, then every natural number must be equal to 1 1. Prove it. b. The result of 2=12 = 1 is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong?
p5. To calculate (1998)(1996)(1994)(1992)+16\sqrt{(1998)(1996)(1994)(1992)+16} . someone does it in a simple way as follows: 20002−2×5×2000+52−52000^2-2 \times 5\times 2000 + 5^2 - 5? Is the way that person can justified? Why?
p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than 25,00025,000 Rp.. Behind every coupon is written one of the following numbers: 99, 1212, 4242, 5757, 6969, 2121, 15, 7575, 2424 and 8181. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV 21′′21''. If the restaurant owner provides as much as 1010 21′′21'' TV pieces, how many should be handed over to the the customer?
p7. Given is the shape of the image below. https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png The centers of circles BB, CC, DD, and EE are placed on the diameter of circle AA and the diameter of circle BB is the same as the radius of circle AA. Circles CC, DD, and EE are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle AA. What is the ratio of the circumference of the circle AA with the sum of the circumferences of circles BB, CC, DD, and EE?
p8. It is known that a+b+c=0a + b + c = 0. Prove that a3+b3+c3=3abca^3 + b^3 + c^3 = 3abc.
algebrageometrycombinatoricsnumber theoryindonesia juniors