At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges? [i]Ray Li[/i]

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Determine all integers $n\geq 1$ for which the numbers $1,2,\ldots,n$ may be (re)ordered as $a_1,a_2,\ldots,a_n$ in such a way that the average $\dfrac {a_1+a_2+\cdots + a_k} {k}$ is an integer for all values $1\leq k\leq n$. (Dan Schwarz)

Find all pairs $(p,q)$ of prime numbers such that $$p^q-4~|~q^p-1.$$ [i](Vlad Robu)[/i]

a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases} a_1 - 3a_2 + 2a_3 \ge 0, \\ a_2 - 3a_3 + 2a_4 \ge 0, \\ a_3 - 3a_4 + 2a_5 \ge 0, \\ ... \\ a_{99} - 3a_{100} + 2a_1 \ge 0, \\ a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$ prove that the numbers are equal. b) Given numbers $a_1=1, ..., a_{100}$ such that $\begin{cases} a_1 - 4a_2 + 3a_3 \ge 0, \\ a_2 - 4a_3 + 3a_4 \ge 0, \\ a_3 - 4a_4 + 3a_5 \ge 0, \\ ... \\ a_{99} - 4a_{100} + 3a_1 \ge 0, \\ a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$ Find $a_2, a_3, ... , a_{100}.$

A tourist is learning an incorrect way to sort a permutation $(p_1, \dots, p_n)$ of the integers $(1, \dots, n)$. We define a [i]fix[/i] on two adjacent elements $p_i$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_i>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fixes, numbered $a=1, 2, \dots, n-1$. In round $a$ of fixes, the tourist fixes $p_a$ and $p_{a+1}$, then $p_{a+1}$ and $p_{a+2}$, and so on, up to $p_{n-1}$ and $p_n$. In this process, there are $(n-1)+(n-2)+\dots+1 = \tfrac{n(n-1)}{2}$ total fixes performed. How many permutations of $(1, \dots, 2018)$ can the tourist start with to obtain $(1, \dots, 2018)$ after performing these steps?

Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$.

In Genshin Impact, $PRIMOGEM'$ is the octagon in the diagram below. Let $A$ be the intersection of $PO$ and $IE$. Suppose $PR=RI=IM=MO=OG=GE=EM'=M'P$, $AP=AI=AO=AE=4$, and $AR=AM=AG=AM'=\sqrt{2}$. Find the area of $PRIMOGEM'$. [asy] size(5cm); pair P = (0, 4), R = (1, 1), I = (4, 0), M = (1, -1), O = (0, -4), G = (-1, -1), E = (-4, 0), MM = (-1, 1), origin = (0, 0); draw(P--R--I--M--O--G--E--MM--P); draw(origin--P); draw(origin--I); draw(origin--O); draw(origin--E); draw(R--G); draw(MM--M); label("$P$", P, N); label("$R$", R, NE); label("$I$", I, SE); label("$M$", M, SE); label("$G$", G, SW); label("$E$", E, W); label("$M'$", MM, NW); label("$O$", O, S); [/asy]

Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

Maria was a brilliant mathematician who found the following property about her year of birth: if $f$ is a function defined in the set of natural numbers $N = \{0, 1, 2, 3, 4, 5,...\}$ such that $f(1) = 1335$ and $f(n+1) = f(n)-2n+43$ for all $n \in N$, then his year of birth is the maximum value that $f(n)$ can reach when $n$ takes values in $N$. Determine the year of birth of Mary.

Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\dfrac{BP}{CP} = 3$. Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$, $n$, $p$, $q$ with $\gcd(m,n) = \gcd(p,q) = 1$. What is $m+n+p+q$? $\phantom{}$ $\textbf{(A) } 42 \qquad \textbf{(B) }86 \qquad \textbf{(C) } 92 \qquad \textbf{(D) } 114 \qquad \textbf{(E) } 130$

A ring is of the shape of a hoola-hoop of negligible thickness. A ring of radius $R$ carries a current $I$. Prove that the magnetic field at a given point in the plane of the ring at a distance $a$ from the center, due to the magnetic field of the ring, is \[ B = \dfrac {\mu_0}{2\pi} \cdot IR \cdot \displaystyle\int_{0}^{\pi} \dfrac {R - a \cos \theta}{\sqrt{\left( a^2 + R^2 - 2aR \cos \theta \right)^3}} \, \mathrm{d}\theta. \] [i]Problem proposed by Ahaan Rungta[/i]