Let $x, y$, and $z$ be real numbers satisfying $\begin{aligned} & 4\left(x^2+y^2+z^2\right)=a \\ & 4(x-y-z)=3+a \end{aligned}$ Then $a$ equals

Consider two sets $A=\{2,3,5,7,11,13\}$ and $B=\{1,8,27\}$ . Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$ , there is at least one element $a$ in $A$ such that $f(a)=b$ . Then, the total number of such functions $f$ is

A function $f$ maps the set of natural numbers to whole numbers, such that $f(x y)=f(x) f(y)+f(x)+f(y)$ for all $x, y$ and $f(p)=1$ for every prime number $p$ . Then, the value of $f(160000)$ is

For any non-zero real number $x$ , let $f(x)+2 f\left(\frac{1}{x}\right)=3 x$ . Then, the sum of all possible values of $x$ for which $f(x)=3$ , is

Suppose $f(x, y)$ is a real-valued function such that $f(3 x+2 y, 2 x-5 y)=19 x$ , for all real numbers $x$ and $y$ . The value of $x$ for which $f(x, 2 x)=27$ , is

Let $0 \leq a \leq x \leq 100$ and $f(x)=|x-a|+|x-100|+|x-a-50|$ . Then the maximum value of $f(x)$ becomes 100 when $a$ is equal to

Suppose for all integers $x$, there are two functions $f$ and $g$ such that $f(x) + f(x-1) - 1 = 0$ and $g(x) = x^2$. If $f\left(x^2 - x\right) = 5$, then the value of the sum $f(g(5)) + g(f(5))$ is

Find the minimum value of $,\dfrac{x^2 - 6x + 10}{3 - x},$ for $x < 3$

Let $r$ be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.$ . Then, the equation $f(x)=f(f(x))$ holds for all real values of $x$ where

If x₀ = 1, x₁ = 2, and $x_{n+2} = \frac{1 + x_{n+1}}{x_n}$, $n = 0, 1, 2, 3, \ldots$, then $x_{2021}$ is equal to?

f(x) = $\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ is negative if and only if

For all real values of x, the range of the function f(x) = $\frac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}$ is

If $f(x)=x^{2}-7 x$ and $g(x)=x+3$ , then the minimum value of $f(g(x))-3 x$ is

The number of real-valued solutions of the equation $2^x + 2^{-x} = 2 - (x - 2)^2$ is

If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

For real $x$, the maximum possible value of $\frac{x}{\sqrt{1 + x^{4}}}$ is

Let $k$ be a constant. The equations $kx + y = 3$ and $4x + ky = 4$ have a unique solution if and only if

If f(x+y) = f(x)f(y) and f(5) = 4, then f(10) - f(-10) is equal to

If x₁ = -1 and xₘ = xₘ₊₁ + (m + 1) for every positive integer m, then x₁₀₀ equals

The number of the real roots of the equation 2cos(x(x + 1)) = 2 x + 2 -x is

Consider a function $f(x+y) = f(x) \times f(y)$ where $x$, $y$ are positive integers, and $f(1) = 2$. If $f(a+1) + f(a+2) + \text{...} + f(a+n) = 16 (2^n - 1)$ then $a$ is equal to.

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals [TITA]

Let f be a function such that f(mn) = f(m) f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals [TITA]

Let $f(x) = \min\{2x^2,\ 52 - 5x\}$ where $x$ is a positive real number. Then maximum possible value of $f(x)$?

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals. [TITA]

Let f(x)=min{2x2, 52 - 5x}, where x is any positive real number. Then the maximum possible value of f(x) is [TITA]

Let $f(x) = \max\{5x, 52 - 2x^2\}$, where $x$ is any positive real number. Then the minimum possible value of $f(x)$ is (TITA).

If $f(x) = \dfrac{5x + 2}{3x - 5}$ and $g(x) = x^2 - 2x - 1$, then the value of $g(f(f(3)))$ is:

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is [TITA]

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

Let $f(x) = x^2$ and $g(x) = 2x$, for all real $x$. Then the value of $f(f(g(x)) + g(f(x)))$ at $x = 1$ is