Let and Find the minimum

Let be integer numbers satisfy that:. Prove that:
a) are even
b) is a perfect square number
c) can’t be any power of a positive integer number

is given where are positive reals. Prove that:

Let be a trapezoid inscribed in circle , . Let is the symmetric point of across , intersects at . Let is midpoint of and is orthocenter of triangle . On take a point so that . Prove that: coincide

Determine the largest two-digit number with the following property:
for any six-digit number number is a divisor of the number if and only if the number is a divisor of the corresponding three-digit number .

Note The numbers and need not be different.

Find all functions satisfying

for all real numbers and , with the additional constraint .

You may or may have not noticed that the tag for the ADFGX cipher of the previous post was ADFGVX. This is becuase the ADFGX cipher is actually a previous version of this cipher, the ADFGVX cipher. The new cipher has an extra letter (the V in case you hadn't noticed). This leaves 36 spaces in the cipher grid instead of 25. This is exactly enough space for the 26 letters of the alphabet and 10 digits. This would make statistics like temperature and # of dead in wartime settings much easier to convey. The process of enciphering and deciphering are exactly the same. Here is a sample grid:

_ A D F G V X
A q w 3 r t e
D y m u 7 i d
F 4 2 x c b n
G l h j 8 2 a
V 0 9 5 d g f
X z k L o p r

*Note: I changed the lowercase l to an uppercase L to distinguish between the l and the one. They'd look really similar otherwise (1, l)

During WWI, the Germans used this cipher frequently. It worked like this:

_ A D F G X
A t h e q u
D i c k b r
F o w n f x
G m p s v a
X l z y d g

We create a 5 by 5 matrix like we did with the playfair cipher, again, we omit the letter J, when encrypting your ciphertext convert all J's to I's. As you may have guessed, every letter has a digraph mapped to it, for example, the letter "s" is equal to "GF" because it is on row G and column F. The letters A, D, F, G, and X were chosen because they sound very different when they were written in Morse Code so that there would be no errors in transmission if the ciphertext was transmitted via a telegraph. We also have to choose another keyword, for this example we will choose the word "LAZY".

Here is our plaintext:
"seven oceans on the planet earth".
We first remove all spaces:
"sevenoceansontheplanetearth"
then, we convert each letter into it's corresponding digraph.

"GF AF GG AF FF FA DD AF GX FF GF FA FF AA AD AF GD XA GX FF AF AA AF GX DX AA AD"

Now we remove spaces again:
"GFAFGGAFFFFADDAFGXFFGFFAFFAAADAFGDXAGXFFAFAAAFGXDXAAAD"

Forget about this for a sec and go back to our second keyword "LAZY", write it down:

L A Z Y

Now put the string I told you to forget about under the 4 letters wrapping about like this:

L A Z Y
G F A F
G G A F
F F F A
D D A F
G X F F
G F F A
F F A A
A D A F
G D X A
G X F F
A F A A
A F G X
D X A A
A D F D

I filled the remaining two spaces next to the last 2 characters with two "null" characters that mean nothing.

Next, alphabetize the 2nd keyword: LAZY --> ALYZ. When you move the letters of the keyword around to alphabetize it, also move the columns under the letter around so that the column under the A is still under the A after the alphabetization and the column under the L is still under the L after the alphabetization and so on:

A L Y Z
f g f a
g g f a
f f a f
d d f a
x g f f
f g a f
f f a a
d a f a
d g a x
x g f f
f a a a
f a x g
x d a a
d a d f

Then you read out the resulting letters by row: fgfaggfaffafddfaxgfffgafffaadafadgaxxgfffaaafaxgxdaadadf.

This is your ciphertext, to decrypt reverse the process and ignore any letters at the end that do not make sense, these letters are from the null characters you added.

To recognize that a cipher is an ADFGX cipher, if the ciphertext has only the letters A, D, F, G, and X, then it is probably an ADFGX cipher.

Let , , , be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, . Prove that for all .

Proposed by Morteza Saghafian, Iran

Let be positive odd integers. Prove that doesn't divide .