Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.

Let $S$ be a set of positive integers such that for every $a,b\in S$, there always exists $c\in S$ such that $c^2$ divides $a(a+b)$. Show that there exists an $a\in S$ such that $a$ divides every element of $S$. [i]Proposed by usjl[/i]

Inside triangle \(ABC\), point \(P\) is marked. Point \(Q\) is on segment \(AB\), and point \(R\) is on segment \(AC\) such that the circumcircles of triangles \(BPQ\) and \(CPR\) are tangent to line \(AP\). Lines are drawn through points \(B\) and \(C\) passing through the center of the circumcircle of triangle \(BPC\), and through points \(Q\) and \(R\) passing through the center of the circumcircle of triangle \(PQR\). Prove that there exists a circle tangent to all four drawn lines.

$N$ oligarchs built a country with $N$ cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is $1$ (note that there can be more than $1$ road going through two cities, but they would belong to different oligarchs). A total of $d$ roads were built. Some oligarchs wanted to create a corporation by combining their cities and roads so that from any city of the corporation you can go to any city of the corporation using only corporation roads (roads can go to other cities outside corporation) but it turned out that no group of less than $N$ oligarchs can create a corporation. What is the maximal amount that $d$ can have?

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?

[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $ [b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $

In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$ and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$. Determine the other two angles of the quadrilateral $ABCD$. (Maria Rozhkova)

Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.

Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

Let $ABCD$ be a quadrilateral, let $P$ be the intersection of $AB$ and $CD$, and let $O$ be the intersection of the perpendicular bisectors of $AB$ and $CD$. Suppose that $O$ does not lie on line $AB$ and $O$ does not lie on line $CD$. Let $B'$ and $D'$ be the reflections of $B$ and $D$ across $OP$. Show that if $AB'$ and $CD'$ meet on $OP$, then $ABCD$ is cyclic.

Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Line $PQ$ intersects $\omega$ at $X$ and $Y$. Assume that $AP=5$, $PB=3$, $XY=11$, and $PQ^2 = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

Find all pairs $(a,b)$ of positive integers satisfying the following conditions: - $a\leq b$ - $ab$ is a perfect cube - No divisor of $a$ or $b$ is a perfect cube greater than $1$ - $a^2+b^2=85\text{lcm}(a,b)$ [i]2020 CCA Math Bonanza Individual Round #12[/i]

Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$. [i]Kvant Magazine [/i]

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: \[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]