U find a 6 digit lock at the bottom o the sea and u must unlock it to get ur oxygen since u r running out of oxygen.
The 6-digit number that you are looking for is the number 10000X+100Y+Z, where X, Y, and Z are integers between 0 and 99 inclusive.
Clue for X: Let AB be the diameter of circle O. Let D be a point outside the circle, such that BD intersects the circle at B and C, and BC=CD. Let AD intersect the circle at A and E. Given that DE=3 and tan(ABE) = 3/sqrt(11), then a possible value of BC^2 can be expressed as a/b, where a and b are relatively prime positive integers not equal to 1. X is equal to the sum of a and b.
Clue for Y: Let a=1, b=2, ..., z=26, and let the value of each word be the sum of the values of its letters. Find the value of the five-letter word which is the source of cashews. (which I have figured out is a "fruit")
There is no clue for Z, but you are also given that all of the digits in the combination are unique, and that the number you are looking for has 8 not necessarily distinct prime factors. From this you should be able to uniquely determine Z.
+9665 Oops sorry wrong answer :( Had to do it all over again.
Just realized angle AEB = angle DEB = 90 degrees.
BD = \(\sqrt{3^2 + \sqrt{11}^2} = \sqrt{20}\)
BC = \(\frac{5\sqrt{2}}{2}=\frac{5}{\sqrt{2}}\)
BC2 = \(\frac{25}{2}\)
X = 25+2 = 27
For Y, I admit that I am wrong and it should be a fruit which is cashew apple which makes 1 + 16 + 16 + 12 + 5 = 50
Now testing numbers between 275011 and 275098
Why 275011 and 275098? It is because the digits can't repeat and 275000 to 275010 and 275099 does not satisfy the conditions.
No such numbers satisfys such conditions.
Lesson taught: Don't even try that lock and quickly get back to water surface 
+9665 X is hard to solve but you can draw a perpendicular line from E to meet the diameter of the circle and name the intersection point F. Then you can prove that triangle BEF is congruent to triangle BED. Then BD = \(\sqrt{11}\) and BC2= \((\frac{\sqrt{11}}{2})^2 = \frac{11}{4}\)
Therefore X = 11+4 = 15.
For Y, the source of cashew. I don't think it's a fruit but I think it's a country. India? China? we will figure out.
India = 9 + 14 + 4 + 9 + 1 = 37
China = 3 + 8 + 9 + 14 + 1 = 35
As the clue of Z stated that all of the digits in the combination are unique. So we'll use India which is 37.
For Z, 8 not necessarily distinct prime factors for all digits...... the hardest of all. Now the number's 1537**
Using a website factorizing numbers, the number is 153720 = 23 × 32 × 5 × 7 × 61
+9665 Oops sorry wrong answer :( Had to do it all over again.
Just realized angle AEB = angle DEB = 90 degrees.
BD = \(\sqrt{3^2 + \sqrt{11}^2} = \sqrt{20}\)
BC = \(\frac{5\sqrt{2}}{2}=\frac{5}{\sqrt{2}}\)
BC2 = \(\frac{25}{2}\)
X = 25+2 = 27
For Y, I admit that I am wrong and it should be a fruit which is cashew apple which makes 1 + 16 + 16 + 12 + 5 = 50
Now testing numbers between 275011 and 275098
Why 275011 and 275098? It is because the digits can't repeat and 275000 to 275010 and 275099 does not satisfy the conditions.
No such numbers satisfys such conditions.
Lesson taught: Don't even try that lock and quickly get back to water surface
