\(p,q\) 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 \(f(f(x))=x.\)

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?

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 \[\sin(mx)+\sin(nx)=2.\]

Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.

Find the number of ways \(66\) identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.

Find the number of positive integers less than \(1000\) that can be expressed as the difference of two integral powers of \(2.\)

Zou and Chou are practicing their \(100\)-meter sprints by running \(6\) races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is \(\frac23\) if they won the previous race but only \(\frac13\) if they lost the previous race. The probability that Zou will win exactly \(5\) of the \(6\) races is \(\frac mn\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n.\)