Let

• · · · + n. Find the value of T1 + T2 + T3 + · · · + T20. 3.

Let p and q be positive integers such that the equation $x^2$ -p

has two distinct positive integer roots, and the equation $x^2$ -q

also has two distinct positive integer roots. The value of p + q equals

Logarithm system with reciprocal relation. Let x > $y > 1$, xy = 64, and $\log_x y + \log_y x$ = 5 2. What is x -y?

Find the number of integer values of a for which $x^2$ -ax + 16 $x^2$ -6x + 8 ≥0 holds for every real $x$ < 2.

A function f is defined on all non - zero real numbers and satisfies $2f(x)$ + 3 f 1

x for every x ≠ 0. Find the value of f(2) + f 1.

For each natural number k, let ak = 2k. Find the smallest natural number m for which (a1)2($a^2$)3($a^3$)4 · · · (a15)16 < a16a17 · · · a15+m.

If x is a real number greater than 1 satisfying (log₂ x)2 -log₂

, then the product of all possible values of x equals.

A function f is defined on the positive integers and satisfies f(1) = 2,

for every n ≥1. The number of positive integers n with 1 ≤n ≤100 for which f(n) is divisible by 5 is:

Factorisation hidden inside a two-variable Diophantine equation Positive integers m and n, with m < n, satisfy

Find the minimum possible value of m + n.

Let x, y, z be powers of 2 greater than 1, not necessarily distinct. They satisfy $\log_x 64 + \log_y 64 + \log_z 64 = \frac{11}{2}$. If xyz is as small as possible, what is x + y + z?

TITA} Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(x)+2 f(1-x)=x^{2}+3x$ for every real $x$. Find the value of $3 f(5)$.

For positive real numbers x and y,

What is the minimum possible value of $x^2$ + 6x y + 3 + $y^2$ + 6y x + 3?

Let

, defined for x ≠ -1. Denote f(n)(x) the n - fold composition f(f(…f(x)…)) with f applied n times. The value of f(2025)(2) is: