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

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 \(m+n.\)

For \(n \ge 1\) call a finite sequence \((a_1, a_2 \ldots a_n)\) of positive integers progressive if \(a_i < a_{i+1}\) and \(a_i\) divides \(a_{i+1}\) for all \(1 \le i \le n-1\). Find the number of progressive sequences such that the sum of the terms in the sequence is equal to \(360\).

Call a positive integer \(n\) \(k\)-pretty if \(n\) has exactly \(k\) positive divisors and \(n\) is divisible by \(k\). For example, \(18\) is \(6\)-pretty. Let \(S\) be the sum of positive integers less than \(2019\) that are \(20\)-pretty. Find \(\tfrac{S}{20}\).

The polynomial \(f(z)=az^{2018}+bz^{2017}+cz^{2016}\) has real coefficients not exceeding \(2019\), and \(f\left(\tfrac{1+\sqrt{3}i}{2}\right)=2015+2019\sqrt{3}i\). Find the remainder when \(f(1)\) is divided by \(1000\).

Triangle \(\triangle{ABC}\) has side lengths \(AB=120,BC=220\), and \(AC=180\). Lines \(\ell_A,\ell_B\), and \(\ell_C\) are drawn parallel to \(\overline{BC},\overline{AC}\), and \(\overline{AB}\), respectively, such that the intersections of \(\ell_A,\ell_B\), and \(\ell_C\) with the interior of \(\triangle ABC\) are segments of lengths \(55,45\), and \(15\), respectively. Find the perimeter of the triangle whose sides lie on lines \(\ell_A,\ell_B\), and \(\ell_C\).

In a Martian civilization, all logarithms whose bases are not specified as assumed to be base \(b\), for some fixed \(b\ge2\). A Martian student writes down\[3\log(\sqrt{x}\log x)=56\]\[\log_{\log x}(x)=54\]and finds that this system of equations has a single real number solution \(x>1\). Find \(b\).

Four ambassadors and one advisor for each of them are to be seated at a round table with \(12\) chairs numbered in order \(1\) to \(12\). Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are \(N\) ways for the \(8\) people to be seated at the table under these conditions. Find the remainder when \(N\) is divided by \(1000\).

A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is \(\tfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

Find the number of \(7\)-tuples of positive integers \((a,b,c,d,e,f,g)\) that satisfy the following systems of equations: \begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}

Lily pads \(1,2,3,\ldots\) lie in a row on a pond. A frog makes a sequence of jumps starting on pad \(1\). From any pad \(k\) the frog jumps to either pad \(k+1\) or pad \(k+2\) chosen randomly with probability \(\tfrac{1}{2}\) and independently of other jumps. The probability that the frog visits pad \(7\) is \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

There is a unique angle \(\theta\) between \(0^{\circ}\) and \(90^{\circ}\) such that for nonnegative integers \(n\), the value of \(\tan{\left(2^{n}\theta\right)}\) is positive when \(n\) is a multiple of \(3\), and negative otherwise. The degree measure of \(\theta\) is \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime integers. Find \(p+q\).

Let \(f(n)\) and \(g(n)\) be functions satisfying $$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$and $$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$for positive integers \(n\) Find the least positive integer \(n\) such that \(\tfrac{f(n)}{g(n)} = \tfrac{4}{7}\).

Find the least positive integer \(n\) for which \(2^n + 5^n - n\) is a multiple of \(1000\).

Two spheres with radii \(36\) and one sphere with radius \(13\) are each externally tangent to the other two spheres and to two different planes \(\mathcal{P}\) and \(\mathcal{Q}\). The intersection of planes \(\mathcal{P}\) and \(\mathcal{Q}\) is the line \(\ell\). The distance from line \(\ell\) to the point where the sphere with radius \(13\) is tangent to plane \(\mathcal{P}\) is \(\tfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).

Find the number of ordered pairs \((m, n)\) such that \(m\) and \(n\) are positive integers in the set \(\{1, 2, ..., 30\}\)and the greatest common divisor of \(2^m + 1\) and \(2^n - 1\) is not \(1.\)

Let \(a, b, c,\) and \(d\) be real numbers that satisfy the system of equations \begin{align*} a+b&=-3\\ ab+bc+ca&= -4\\ abc+bcd+cda+dab&=14\\ abcd&=30. \end{align*}There exist relatively prime positive integers \(m\) and \(n\) such that $$a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.$$Find \(m + n\).

For any finite set \(S\), let \(|S|\) denote the number of elements in \(S\). FInd the number of ordered pairs \((A,B)\) such that \(A\) and \(B\) are (not necessarily distinct) subsets of \(\{1,2,3,4,5\}\) that satisfy $$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$

For positive real numbers \(s\), let \(\tau(s)\) denote the set of all obtuse triangles that have area \(s\) and two sides with lengths \(4\) and \(10\). The set of all \(s\) for which \(\tau(s)\) is nonempty, but all triangles in \(\tau(s)\) are congruent, is an interval \([a,b)\). Find \(a^2+b^2\).

There are real numbers \(a, b, c,\) and \(d\) such that \(-20\) is a root of \(x^3 + ax + b\) and \(-21\) is a root of \(x^3 + cx^2 + d.\) These two polynomials share a complex root \(m + \sqrt{n} \cdot i,\) where \(m\) and \(n\) are positive integers and \(i = \sqrt{-1}.\) Find \(m+n.\)

Find the number of permutations \(x_1, x_2, x_3, x_4, x_5\) of numbers \(1, 2, 3, 4, 5\) such that the sum of five products $$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$$is divisible by \(3.\)

Let \(S\) be the set of positive integers \(k\) such that the two parabolas$$y=x^2-k~~\text{and}~~x=2(y-20)^2-k$$intersect in four distinct points, and these four points lie on a circle with radius at most \(21\). Find the sum of the least element of \(S\) and the greatest element of \(S\).

Consider the sequence \((a_k)_{k\ge 1}\) of positive rational numbers defined by \(a_1 = \frac{2020}{2021}\) and for \(k\ge 1\), if \(a_k = \frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\), then \[a_{k+1} = \frac{m + 18}{n+19}.\] Determine the sum of all positive integers \(j\) such that the rational number \(a_j\) can be written in the form \(\frac{t}{t+1}\) for some positive integer \(t \).

Find the number of integers \(c\) such that the equation$$\left||20|x|-x^2|-c\right|=21$$has \(12\) distinct real solutions.

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 $$\sin(mx)+\sin(nx)=2.$$