Show that the set $S$ of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{ for any } p, q \in \mathbb N \right\}$) is not the union of finitely many arithmetic progressions.

(a) FInd the number of complex roots of $Z^6 = Z + \bar{Z}$ (b) Find the number of complex solutions of $Z^n = Z + \bar{Z}$ for $n \in \mathbb{Z}^+$

A [i]windmill[/i] is a closed line segment of unit length with a distinguished endpoint, the [i]pivot[/i]. Let $S$ be a finite set of $n$ points such that the distance between any two points of $S$ is greater than $c$. A configuration of $n$ windmills is [i]admissible[/i] if no two windmills intersect and each point of $S$ is used exactly once as a pivot. An admissible configuration of windmills is initially given to Geoff in the plane. In one operation Geoff can rotate any windmill around its pivot, either clockwise or counterclockwise and by any amount, as long as no two windmills intersect during the process. Show that Geoff can reach any other admissible configuration in finitely many operations, where (i) $c = \sqrt 3$, (ii) $c = \sqrt 2$. [i]Proposed by Michael Ren[/i]

It is given that there exist unique integers $m_1,\ldots, m_{100}$ such that \[0\leq m_1 < m_2 < \cdots < m_{100}\quad\text{and}\quad 2018 = \binom{m_1}1 + \binom{m_2}2 + \cdots + \binom{m_{100}}{100}.\] Find $m_1 + m_2 + \cdots + m_{100}$.

In the $xOy$ system consider the lines $d_1\ :\ 2x-y-2=0,\ d_2\ :\ x+y-4=0,\ d_3\ :\ y=2$ and $d_4\ :\ x-4y+3=0$. Find the vertices of the triangles whom medians are $d_1,d_2,d_3$ and $d_4$ is one of their altitudes. [i]Lucian Dragomir[/i]

Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have \[ \left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K. \] [i]2015 CCA Math Bonanza Lightning Round #3.4[/i]

Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$? [i] (Adapted from Archimedes 2011, Traian Preda)[/i]

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to $6$. b) Show, by providing an example, that the number of teams could be equal to $7$.

Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$

For any positive integer $n$, define the subset $S_n$ of natural numbers as follow $$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$ Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$. [i]Proposed by Yahya Motevassel[/i]

Determine all natural numbers $n$ of the form $n=[a,b]+[b,c]+[c,a]$ where $a,b,c$ are positive integers and $[u,v]$ is the least common multiple of the integers $u$ and $v$.

Fencing Ferdinand wants to fence three rectangular areas. there are fences in three types, with $4$ amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?

Let $r, s$, and $t$ be the three roots of the equation $8x^3 + 1001x + 2008 = 0$. Find $(r + s)^3 + (s + t)^3 + (t + r)^3$ .

The area of rectangle $ABCD$ is $72$. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, the area of that triangle is [asy] pair A,B,C,D; A = (0,8); B = (9,8); C = (9,0); D = (0,0); draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); [/asy] $\text{(A)}\ 21 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40$

Points $P$ and $Q$ inside a triangle $ABC$ with sides $a,b,c$ and the corresponding angle $\alpha,\beta,\gamma$ satisfy $\angle BPC=\angle CPA=\angle APB=120^\circ$ and $\angle BQC=60^\circ+\alpha$, $\angle CQA=60^\circ+\beta$, $\angle AQB=60^\circ+\gamma$. Prove the equality $$(AP+BP+CP)^3\cdot AQ\cdot BQ\cdot CQ=(abc)^2.$$

In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. [i]Р. Sadykov[/i]

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.

Through points $ A $ and $ B $ two oblique lines $ m $ and $ n $ are drawn perpendicular to the line $ AB $. On line $ m $ the point $ C $ (different from $ A $) is taken, and on line $ n $ the point $ D $ (different from $ B $) is taken. Given the lengths of segments $ AB = d $ and $ CD = l $ and the angle $ \varphi $ formed by the oblique lines $ m $ and $ n $, calculate the radius of the surface of the sphere passing through the points $ A $, $ B $, $ C $, $ D $.

Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?

[b]Problem 4[/b] Find all continuously differentiable functions $ f : \mathbb{R} \rightarrow \mathbb{R} $, such that for every $a \geq 0$ the following relation holds: $$\iiint \limits_{D(a)} xf \left( \frac{ay}{\sqrt{x^2+y^2}} \right) \ dx \ dy\ dz = \frac{\pi a^3}{8} (f(a) + \sin a -1)\ , $$ where $D(a) = \left\{ (x,y,z)\ :\ x^2+y^2+z^2 \leq a^2\ , \ |y|\leq \frac{x}{\sqrt{3}} \right\}\ .$

The vertical cross section of a circular cone with vertex $P$ is an isoceles right triangle. Point $A$ is on the base circle, point $B$ is interior to the base circle, $O$ is the center of the base circle, $AB \perp OB$ at $B$, $OH \perp PB$ at $H$, $PA = 4$, and $C$ is the midpoint of $PA$. When the volume of tetrahedron $OHPC$ is maximized, the length of $OB$ is $x$. $x^2$ is in the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions: (1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n $; (2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ($i=1,2,\cdots ,n-2$); (3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ($i=1,2,\cdots ,n-2$). Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$.

Let $x$, $y$, and $z$ be real numbers (not necessarily positive) such that $x^4+y^4+z^4+xyz=4$. Show that $x\le2$ and $\sqrt{2-x}\ge\frac{y+z}{2}$. [i]Proposed by Alyazeed Basyoni[/i]