practice problems

Q1.  Let (a,b,c) be distinct non-zero real numbers satisfying a + 2/b = b + 2/c = c + 2/a. Determine the value of |a^2.b + b^2.c + c^2.a|.
S1.
The fact that a,b,c are given as distinct => problem wants us to cancel out factors like a-b,b-c,c-a.
a-b = 2(c-b)/bc similarly create other equations and multiply all to get (abc)^2 = 8
Then again simplify the given equations to get a^2b + ... = 3abc => answer = 6.sqrt(2)

Q2. Let (S) be the set of all three-digit positive integers (abc) (where (a,b,c) are the digits) such that the two-digit number (ab) is divisible by (c), and the two-digit number (bc) is divisible by (a).
If we exclude the trivial cases (a=b=c), what is the largest three-digit integer in set (S)?

S2.
a | bc and c | ab
To make it largest, try a = 9
9 | bc c | 9b
Now let's try values of bc
bc = 99 trivial
bc = 90 c can't be 0 as c | ab fails
bc = 81 works since 1 | 98
So answer = 981

Q3. Find infinitely many triples ((a,b,c)) of positive integers such that (a, b, c) are in arithmetic progression and such that (ab+1), (bc+1), and (ca+1) are perfect squares.
S3.
Prereqs:
Pell's equations
Diophantine m-tuples

If ab + 1 = k^2 then Euler substitution tells us  that c = a + b + 2k will make other 2 terms also squares.
c = 2b- a => b = 2a + 2k
ab + 1 = k^2 => k = a +- sqrt(3a^2 + 1)
for k to be integer 3a^2 + 1 = m^2 which is Pell's equation and has infinitely many solutions.
So we get a => we get k => we get b => we get c. (all infinite)
H.P.