Starting with the sequence $F_1 = (1,2,3,4, \ldots)$ of the natural numbers further sequences are generated as follows: $F_{n+1}$ is created from $F_n$ by the following rule: the order of elements remains unchanged, the elements from $F_n$ which are divisible by $n$ are increased by 1 and the other elements from $F_n$ remain unchanged. Example: $F_2 = (2,3,4,5 \ldots)$ and $F_3 = (3,3,5,5, \ldots)$. Determine all natural numbers $n$ such that exactly the first $n-1$ elements of $F_n$ take the value $n.$

Let $m$ be a positive integers. A square room with corners at $(0,0), (2m,0), (0,2m),$ $(2m,2m)$ has mirrors as walls. At each integer lattice point $(i,j)$ with $0 < i, j < 2m$ a single small double sided mirror is oriented parallel to either the $x$ or $y$ axis. A beam of light is shone from a corner making a $45^\circ$ angle with each of the walls. Prove that the opposite corner is not lit.

[u]Round 1[/u] [b]p1.[/b] Define the operation $\clubsuit$ so that $a \,\clubsuit \, b = a^b + b^a$. Then, if $2 \,\clubsuit \,b = 32$, what is $b$? [b]p2. [/b] A square is changed into a rectangle by increasing two of its sides by $p\%$ and decreasing the two other sides by $p\%$. The area is then reduced by $1\%$. What is the value of $p$? [b]p3.[/b] What is the sum, in degrees, of the internal angles of a heptagon? [b]p4.[/b] How many integers in between $\sqrt{47}$ and $\sqrt{8283}$ are divisible by $7$? [u]Round 2[/u] [b]p5.[/b] Some mutant green turkeys and pink elephants are grazing in a field. Mutant green turkeys have six legs and three heads. Pink elephants have $4$ legs and $1$ head. There are $100$ legs and $37$ heads in the field. How many animals are grazing? [b]p6.[/b] Let $A = (0, 0)$, $B = (6, 8)$, $C = (20, 8)$, $D = (14, 0)$, $E = (21, -10)$, and $F = (7, -10)$. Find the area of the hexagon $ABCDEF$. [b]p7.[/b] In Moscow, three men, Oleg, Igor, and Dima, are questioned on suspicion of stealing Vladimir Putin’s blankie. It is known that each man either always tells the truth or always lies. They make the following statements: (a) Oleg: I am innocent! (b) Igor: Dima stole the blankie! (c) Dima: I am innocent! (d) Igor: I am guilty! (e) Oleg: Yes, Igor is indeed guilty! If exactly one of Oleg, Igor, and Dima is guilty of the theft, who is the thief?? [b]p8.[/b] How many $11$-letter sequences of $E$’s and $M$’s have at least as many $E$’s as $M$’s? [u]Round 3[/u] [b]p9.[/b] John is entering the following summation $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39$ in his calculator. However, he accidently leaves out a plus sign and the answer becomes $3582$. What is the number that comes before the missing plus sign? [b]p10.[/b] Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$? [b]p11.[/b] Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps? [b]p12.[/b] Tracy wants to jump through a line of $12$ tiles on the floor by either jumping onto the next block, or jumping onto the block two steps ahead. An example of a path through the $12$ tiles may be: $1$ step, $2$ steps, $2$ steps, $2$ steps, $1$ step, $2$ steps, $2$ steps. In how many ways can Tracy jump through these $12$ tiles? PS. You should use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784268p24464984]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.

If $-3 \le x < \frac{3}{2}$ and $x \ne 1$, define $C(x) = \frac{x^3}{1 - x}$. The real root of the cubic $2x^3 + 3x - 7$ is of the form $p C^{-1}(q)$, where $p$ and $q$ are rational numbers. What is the ordered pair $(p, q)$?

Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC, CA,$ and $AB$ at $D,E$ and $F$ respectively. If $BD=x, CE=y$ and $AF=z$, then show that \[r^2=\frac{xyz}{x+y+z}\]

The positive integers $p$, $q$ are such that for each real number $x$ $(x+1)^p (x-3)^q=x^n+a_1 x^{n-1}+a_2 x^{n-2}+\dots +a_{n-1} x+a_n$ where $n=p+q$ and $a_1,\dots ,a_n$ are real numbers. Prove that there exists infinitely many pairs $(p,q)$ for which $a_1=a_2$.

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

Let $n$ be an integer. Dayang are given $n$ sticks of lengths $1,2, 3,..., n$. She may connect the sticks at their ends to form longer sticks, but cannot cut them. She wants to use all these sticks to form a square. For example, for $n = 8$, she can make a square of side length $9$ using these connected sticks: $1 + 8$, $2 + 7$, $3 + 6$, and $4 + 5$. How many values of $n$, with $1 \le n \le 2018$, that allow her to do this?

Determine the smallest integer $n$ for which there exist integers $x_1,\ldots,x_n$ and positive integers $a_1,\ldots,a_n$ so that \begin{align*} x_1+\cdots+x_n &=0,\\ a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\ a_1^2x_1+\cdots+a_n^2x_n &<0. \end{align*}

$ x \plus{} 19y \equiv 0 \pmod {23}$ and $ x \plus{} y < 69$. How many pairs of $ (x,y)$ are there in positive integers? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 105 \qquad\textbf{(D)}\ 109 \qquad\textbf{(E)}\ \text{None}$

[b](a)[/b] Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$. Prove that this altitude $DH_4$ intersects all the other three altitudes. [b](b)[/b] If we further know that a second altitude, say the one from vertex A to the face $BCD$, also passes through the orthocenter $H_1$ of $\triangle BCD$, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.

Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.

From a set of integers $\{1,...,100\}$, $k$ integers were deleted. Is it always possible to choose $k$ distinct integers from the remaining set such that their sum is $100$ if [b](a) $k=9$?[/b] [b](b) $k=8$?[/b]

trapezoid has side lengths $10, 10, 10$, and $22$. Each side of the trapezoid is the diameter of a semicircle with the two semicircles on the two parallel sides of the trapezoid facing outside the trapezoid and the other two semicircles facing inside the trapezoid as shown. The region bounded by these four semicircles has area $m + n\pi$, where m and n are positive integers. Find $m + n$. [img]https://3.bp.blogspot.com/-s8BoUPKVUQk/XoEaIYvaz4I/AAAAAAAALl0/ML0klwHogGYWkNhY6maDdI93_GkfL_eyQCK4BGAYYCw/s200/2018%2Bps%2Bhs9.png[/img]

Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle. [i]Alexandr Matveev[/i]

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.

Find all polynomials $P(x)$ and $Q(x)$ with real coefficients such that $$P(Q(x))=P(x)^{2017}$$ for all real numbers $x$.

Given are 10 quadric equations $x^2+a_1x+b_1=0$, $x^2+a_2x+b_2=0$,..., $x^2+a_{10}x+b_{10}=0$. It is known that each of these equations has two distinct real roots and the set of all solutions is ${1,2,...10,-1,-2...,-10}$. Find the minimum value of $b_1+b_2+...+b_{10}$

Complex number $z=\cos\theta+\text{i}\sin\theta(0\leq\theta\leq\pi)$. Points that three complex numbers $z,(1+\text{i})z,2\overline{z}$ refer to on complex plane are $P,Q,R$. When $P,Q,R$ are not collinear, $PQSR$ is a parallelogram. The longest distance between $S$ and the original point is________.

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

A circle centered at $A$ intersects sides $AC$ and $AB$ of $\triangle ABC$ at $E$ and $F$, and the circumcircle of $\triangle ABC$ at $X$ and $Y$. Let $D$ be the point on $BC$ such that $AD$, $BE$, $CF$ concur. Let $P=XE\cap YF$ and $Q=XF\cap YE$. Prove that the foot of the perpendicular from $D$ to $EF$ lies on $PQ$.

Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$.