Given that the integers \(a, b\) satisfy the equation \[\left[\frac{\frac{1}{a}}{\frac{1}{a}-\frac{1}{b}}-\frac{\frac{1}{b}}{\frac{1}{a}+\frac{1}{b}}\right]\left({\frac{1}{a}-\frac{1}{b}}\right)\cdot\frac{1}{\frac{1}{a^2}+\frac{1}{b^2}}=\frac{2}{3},\]find the value of \(a+b.\)

\(p,q\) are two integers, and the two roots of the equation in \(x\) \[x^2-\frac{p^2+11}{9}{x}+\frac{15}{4}{(p+q)}+16=0\] are \(p\) and \(q\) also. Find the values of \(p\) and \(q.\)

Let \(p\) be a positive prime number such that the equation \[x^2-px-580p=0\] has two integer solutions. Find the value of \(p.\)

How many ordered pairs of integers \((x,y)\) satisfy the equation \[x^2+y^2=2(x+y)+xy?\]

Find the number of positive integer solutions of the equation \[\frac{2}{x}-\frac{3}{y}=\frac{1}{4}.\]

Find the number of positive integer solutions to the equation \[\frac{x}{3}+\frac{14}{y}=3.\]

Find the number of non-zero integer solutions \(x,y\) to the equation \[\frac{15}{x^2y}+\frac{3}{xy}-\frac{2}{x}=2.\]

Let \(x, y, z\) and \(w\) represent four \(\textit{distinct}\) positive integers such that \[x^2-y^2=z^2-w^2=81.\] Find the value of \(xz+yw+xw+yz.\)

How many number of pairs \((x,y)\) of two integers satisfy the equation \[x^2-y^2=12?\]

Given that \(\frac{1260}{a^2+a-6}\) is a positive integer, where \(a\) is a positive integer. Find the value of \(a.\)

Define the sequence \(\{a_n\}_{n\geq 1}\) as \(a_n=n-1\), \(n\leq 2\) and \(a_n=\) remainder left by \(a_{n-1}+a_{n-2}\) when divided by \(3\) \(\forall n\geq 2\). Then \[\sum_{i=2018}^{2025}a_i=?\]

Given that the equation \((m^2-12)x^4-8x^2-4=0\) has no real roots, then the largest value of \(m\) is \(p\sqrt{q}\), where \(p\) and \(q\) are natural numbers, \(q\) is square-free. Determine \(p+q\).

Find the number of \(f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}\) such that \(f(f(x))=x.\)

(i) Let \(a_1,a_2,...,a_n\) be n real numbers. Show that there exists some real number \(\alpha\) such that \(a_1+\alpha,a_2+\alpha,...,a_n+\alpha\) are all irrational. (ii) Prove that such a satetement is not valid if all these are rquired to be rational.

Let \(P(x),Q(x)\) be monic polynomials with integer coeeficients. Let \(a_n=n!+n\) for all natural numbers \(n\). Show that if \(\frac{P(a_n)}{Q(a_n)}\) is an integer for all positive integer \(n\) then \(\frac{P(n)}{Q(n)}\) is an integer for every integer \(n\neq0\).

Prove that any integer has a multiple consisting of all ten digits \[\{0,1,2,3,4,5,6,7,8,9\}.\] Note: Any digit can be repeated any number of times

Find all polynomial \(P(x)\) with degree \(\leq n\) and non negative coefficients such that \[P(x)P(\frac{1}{x})\leq P(1)^2\]for all positive \(x\). Here \(n\) is a natuaral number.

Let \(a\) be a fixed real number. Consider the equation \[(x+2)^{2}(x+7)^{2}+a=0, x \in R\] where \(R\) is the set of real numbers. For what values of \(a\), will the equation have exactly one double-root?

Let \(i\) be a root of the equation \(x^2+1=0\) and let \(\omega\) be a root of the equation \(x^2+x+1=0\) . Construct a polynomial\[f(x)=a_0+a_1x+\cdots+a_nx^n\]where \(a_0,a_1,\cdots,a_n\) are all integers such that \(f(i+\omega)=0\).

Find the number of pairs \((m,n)\) of positive integers with \(1 \le m < n \le 30\) such that there exists a real number \(x\) satisfying \[\sin(mx)+\sin(nx)=2.\]