▪ Japan Mathematical Olympiad Preliminary 2012

09 January 2012

1. Given a square . Let    

and  and let . If 

 , 

then find the side length of the square. 

2. In the attached figure as below, given are in a circle in this order and suppose the angle of the tangent at and the line is , and that of the tangent at and the line is . If and they are located across the center of the circle, then find . 

3. Let be distinct integers. Denote by the maximum value of . Find the possible minimum value of . 

4. Let be a positive integer which is a multiple of 3, but isn't a multiple of 9. If adding the product of each digit of to gives a multiple of 9, then find the possible minimum value of . 

5. Find all positive integers such that the product of all positive divisors of is . 

6. Given a grid , each grid square is colored either red or blue, in such a way that :

There exists at least one grid square colored red, and at least one colored blue.

All grid squares colored red are connected, and so are all grid squares colored blue.

How many possibilities are there for such colorings?

Note that two distinct grid squares are considered to be connected when they share an edge. 

7. Given a triangle with circumcenter . Let such that and . If is the midpoint of , then find . 

8. How many methods to write down the array of the positive integers such that :

Write down in the first place, in the last .

Note : The next to is written a positive number which is less than . 

9. and wrote two positive integers respectively in a black board. The product of the numbers written by was twice as the sum of the numbers written by and the product of the numbers written by was twice as the sum of the numbers written by and the sum of the numbers written by is more than or equal to that of . Find all possible sum written by .

Note: Four numbers written are not always distinct. 

10. How many positive integers are there such that

 

11. Let be a positive integer. Given a grid by , color exactly squares satisfying the condition 

that :

if a given square is colored, then the other squares that share only a single vertex (i.e., not a whole edge) with that square aren't colored.

How many possibilities are there coloring in such a way?

Note that two colorings are considered to be distinct if they differ by a rotation or reflection of the square.

12. In a plane given are a cube with sidelength and a plane, the common part of the cubic and the plane form 6-polygon such that 

.

If 

,

then find the value of 

.