For each pair of positive integers $m$ and $n$, we define $f_m(n)$ as follows: $$ f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k), $$ where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$. For example, $f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$. $a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$. $b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$.

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? $\textbf{(A)}\hspace{.05in}3127 \qquad \textbf{(B)}\hspace{.05in}3133 \qquad \textbf{(C)}\hspace{.05in}3137 \qquad \textbf{(D)}\hspace{.05in}3139 \qquad \textbf{(E)}\hspace{.05in}3149 $

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? $\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.

Are there infinitely many positive integers that [b]can’t[/b] be presented as a sum of no more than fifteen fourth degrees of positive integers. (For example 15 isn’t such number as it can be presented as the sum of $15.1^4$)

Let $M$ be a point inside square $ABCD$ and $A_1,B_1,C_1,D_1$ be the second intersection points of $AM$, $BM$, $CM$, $DM$ with the circumcircle of the square. Prove that $A_1B_1\cdot C_1D_1=A_1D_1\cdot B_1C_1$.

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

Alice and Billy are playing a game on a number line. They both start at $0$. Each turn, Alice has a $\frac{1}{2}$ chance of moving $1$ unit in the positive direction, and a $\frac{1}{2}$ chance of moving $1$ unit in the negative direction, while Billy has a $\frac{2}{3}$ chance of moving $1$ unit in the positive direction, and a $\frac{1}{3}$ chance of moving $1$ unit in the negative direction. Alice and Billy alternate turns, with Alice going first. If a player reaches $2$, they win and the game ends, but if they reach $-2$, they lose and the other player wins, and the game ends. What is the probability that Billy wins? [i]2018 CCA Math Bonanza Lightning Round #4.4[/i]

$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$ $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$

Find all integers $ (x,y,z)$, satisfying equality: $ x^2(y \minus{} z) \plus{} y^2(z \minus{} x) \plus{} z^2(x \minus{} y) \equal{} 2$

Let $n$ be a positive integer. We want to make up a collection of cards with the following properties: 1. each card has a number of the form $m!$ written on it, where $m$ is a positive integer. 2. for any positive integer $ t \le n!$, we can select some card(s) from this collection such that the sum of the number(s) on the selected card(s) is $t$. Determine the smallest possible number of cards needed in this collection.

Prove that if the relationship between the sides and opposite angles $ A $ and $ B $ of the triangle $ ABC $ is $$ (a^2 + b^2) \sin (A - B) = (a^2 - b^2) \sin (A + B)$$ then such a triangle is right-angled or isosceles.

Find the number of squares such that the sides of the square are segments in the following diagram and where the dot is inside the square. [center][img]https://snag.gy/qXBIY4.jpg[/img][/center]

If a worker receives a $ 20$ percent cut in wages, he may regain his original pay exactly by obtaining a raise of: $ \textbf{(A)}\ \text{20 percent} \qquad \textbf{(B)}\ \text{25 percent} \qquad \textbf{(C)}\ 22\frac{1}{2} \text{ percent} \qquad \textbf{(D)}\ \$20 \qquad \textbf{(E)}\ \$25$

In a triangle $\Delta ABC$, $|AB|=|AC|$. Let $M$ be on the minor arc $AC$ of the circumcircle of $\Delta ABC$ different than $A$ and $C$. Let $BM$ and $AC$ meet at $E$ and the bisector of $\angle BMC$ and $BC$ meet at $F$ such that $\angle AFB=\angle CFE$. Prove that the triangle $\Delta ABC$ is equilateral.

Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2 $ times as large as the original number.

Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is congruent to $\triangle H_AH_BH_C$.

Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$. Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar to triangle $ABC$.

Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Let $p(x) = a_{21} x^{21} + a_{20} x^{20} + \dots + a_1 x + 1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one of its factors are less than $a$. [i]Submitted by Navid Safaei, Tehran, Iran[/i]

Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that: $ f(x)+f(\frac1x)=1$ $ 2f(f(x))=f(2x)$

Positive integers $(p,a,b,c)$ called [i]good quadruple[/i] if a) $p $ is odd prime, b) $a,b,c $ are distinct , c) $ab+1,bc+1$ and $ca+1$ are divisible by $p $. Prove that for all good quadruple $p+2\le \frac {a+b+c}{3} $, and show the equality case.

The $n$ players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left. The players will update their votes via a series of rounds. In one round, each player $a$ updates their vote, one at a time, according to the following procedure: At the time of the update, if $a$ is voting for $b,$ and $b$ is voting for $c,$ then $a$ updates their vote to $c.$ (Note that $a, b,$ and $c$ need not be distinct; if $b=c$ then $a$'s vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds). They repeat this updating procedure for $n$ rounds. Prove that at this time, all $n$ players will unanimously vote for the same person.

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]