practice problems pending

Q1. Find the number of integer solutions for

| |x| - 2020| < 5
S1.
|x| - 2020 = y
=> -5 < y < 5
=> -5 < |x| - 2020 < 5

Part 1:
-5 < |x| - 2020 => 2015 < |x| => x < -2015 or x > 2015

AND
Part 2:
|x| - 2020 < 5
=> |x| < 2025
=> -2025 < x < 2025

Combine both to get x = -2024... -2016 and x = 2016 to 2024
Answer: 18

Q2. Find the sum of positive integers (n) for which

|2^n + 5^n - 65| is a perfect square.
S2.
for n = 1, y = 58
for n = 2, y = 36 = one solution found, n = 2
for n >= 3 y > 0 so we can remove absolute sign.

2^n + 5^n - 65 = k^2
Take mod 3
(-1)^n + (-1)^n = -1 + k^2 mod 3
If n is odd:
-1 = 2 = k^2 mod 3
But k^2 mod 3 can be only 0,1.
=> n has to be even
Let n = 2m and m>=2
y = 4^m + 25^m - 65
m = 2 => y = 16 + 625 - 65 = 576 = 24^2 => n = 4 is a solution.
m = 3 => 4^3 + 25^3 - 65 = k^2 = 25^3 - 1 = 125^2 - 1 so not a square since you won't get 124^2 from here.
For m>=4
=> 4^m - 65 > 0
=> k^2 = 4^m + 25^m - 65 => k^2 > 25^m => k^2 > (5^m)^2
Take the next integer: 5^m + 1
(5^m + 1)^2 = 25^m + 1 + 2.5^m > 25^m + 4^m - 65 = k^2
=> (5^m)^2 < k^2 < (5^m + 1)^2
So k^2 can't be square of an integer.

So there are only 2 solutions: n = 2,4

Q3.

A number (N) in base 10, (503) in base (b), (305) in base (b+2). What is (N)?

S3.
N = 5b^2 + 3 = 3(b+2)^2 + 5 => 2b^2 - 12b - 14 = 0 => b^2 - 6b - 7 = 0 => (b-7)(b+1) = 0 => b = 7
=> N = 5*49 + 3 = 248

Q4.
Two cars initially 500m apart. Car 1 is moving right towards car 2 at 35m/h. Car 2 is moving down south at 50m/h. At the 3 hour mark, the distance between them is increasing or decreasing?
S4.
Car 1 initially at (-500,0), car 2 at (0,0)
Moving along x axis, y axis respectively.
Distance between them is z.
x,0 is the co-ordinate of car 1 and 0,y is of car 2.

x^2 + y^2 = z^2
2xx' + 2yy' = 2zz'
=> xx' + yy' = zz'
At 3 hour mark, x = -395, y = 150, x' = +35, y' = -50, z = sqrt(x^2 + y^2) = 422.52
=> -395*35 -50*150 = 422.5 * z' => z' = negative
z' < 0 => distance is decreasing.

Note that even if you flip the co-ordinate system or choose another origin, result will be the same.

Q5. Given an odd integer (a), prove that

a^2 + (a+2)^2 + (a+4)^2 + 1 is divisible by (12).

S5.

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