Can anyone help with these easy, medium and hard questions?
(1) How many ways are there of placing a single rectangle on this grid so that it completely covers 3 grid squares?
(2) A number is a palindrome if it reads the same as forwards as backwards. The number 131131 is a palindrome; also its first pair of digits, middle and the last pair are prime numbers. How many such 6-digit palindromes are there?
(3) I have a large number of toy soldiers, which I can arrange into a rectangle array consisting of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangle with 5 fewer rows and 5 more columns.
How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?
(4) How many 3 digit numbers are there such that it is equal to the sum of its digits multiplied by 13?
(5) 1 Googol is whereas 1 Googleplex is . Let be the largest whole number for which How many digits does have?
(6) There is a square tabletop filled with small square-shaped tiles. The sum of the squares that form the 2 diagonals is 25. Find the area of the tabletop.
(7) I have a cube which has a surface area of . I slice it into 2 to get 2 (different or same) prisms. (The diagram shows how it has been cut). If one of the prism's surface area is of , then find the surface area of the other prism in relation to .
(8) This is a right-angled triangle. Find the ratio between the unshaded region to the shaded region using the letters in all cases. . Note the blue colour shaded is a square.
(9) A quadrilateral has two parallel sides measuring 25 cm and 37 cm. What is the distance in centimetres, between the midpoints of the diagonals?
That's all for today, folks!
I also wanted to make note of Q5; I myself tried it and came till here; if anyone can help so tell me in the comments.
and 1 Googlepex is = .
To find the number of digits of , we have to find an exponential representation of . Then we can use the theorem:
For any positive integer , the number of digits in is . (It can be found on this wiki )
So, I tried to use the inequality:
First I converted into an equation: and then I didn't understand how to subject .
These are what I attempted:
But I didn't receive any help. Please help me.
I might be adding some more problems in this note this week only so please be updated. I hope to get answers with extremely clear language and makes notes of all theorems used in solving with each method presented carefully and neatly. Please make use of Latex in your answers. I thank everyone who tried.