MathDB

day 1

Part of 2007 Indonesia Juniors

Problems(1)

Indonesia Juniors 2007 day 1 OSN SMP

Source:

10/31/2021
p1. A set of cards contains 100100 cards, each of which is written with a number from 11 up to 100100. On each of the two sides of the card the same number is written, side one is red and the other is green. First of all Leny arranges all the cards with red writing face up. Then Leny did the following three steps: I. Turn over all cards whose numbers are divisible by 22 II. Turn over all the cards whose numbers are divisible by 33 III. Turning over all the cards whose numbers are divisible by 55, but didn't turn over all cards whose numbers are divisible by 55 and 22. Find the number of Leny cards now numbered in red and face up,
p2. Find the area of ​​three intersecting semicircles as shown in the following image. https://cdn.artofproblemsolving.com/attachments/f/b/470c4d2b84435843975a0664fad5fee4a088d5.png
p3. It is known that x+1x=7x+\frac{1}{x}=7 . Determine the value of AA so that Axx4+x2+1=56\frac{Ax}{x^4+x^2+1}=\frac56.
p4. There are 1313 different gifts that will all be distributed to Ami, Ima, Mai,and Mia. If Ami gets at least 44 gifts, Ima and Mai respectively got at least 33 gifts, and Mia got at least 22 gifts, how many possible gift arrangements are there?
p5. A natural number is called a quaprimal number if it satisfies all four following conditions: i. Does not contain zeros. ii. The digits compiling the number are different. iii. The first number and the last number are prime numbers or squares of an integer. iv. Each pair of consecutive numbers forms a prime number or square of an integer.
For example, we check the number 971643971643. (i) 971643971643 does not contain zeros. (ii) The digits who compile 971643971643 are different. (iii) One first number and one last number of 971643971643, namely 99 and 33 is a prime number or a square of an integer. (iv) Each pair of consecutive numbers, namely 97,71,16,6497, 71, 16, 64, and 4343 form prime number or square of an integer. So 971643971643 is a quadratic number.
Find the largest 66-digit quaprimal number. Find the smallest 66-digit quaprimal number. Which digit is never contained in any arbitrary quaprimal number? Explain.
algebrageometrycombinatoricsnumber theoryindonesia juniors