How many subsets of the set $\{1,2,3,4,5,6,7,8,9,10\}$ are there that does not contain 4 consequtive integers? $ \textbf{(A)}\ 596 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 679 \qquad \textbf{(D)}\ 773 \qquad \textbf{(E)}\ 812$

Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)--(1,0)--(0,-2)--(-1,0)--cycle); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),SE); label("$(0,-2)$",(0,-2),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(A)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)..(1,0)..(0,-2)^^(0,-2)..(-1,0)..(0,2)); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),SE); label("$(0,-2)$",(0,-2),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(B)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)--(1,0)--(-1,0)--cycle); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),S); label("$(-1,0)$",(-1,0),S); label("$\textbf{(C)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)..(1/2,3/2)..(1,0)--(-1,0)..(-1/2,3/2)..(0,2)); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),S); label("$(-1,0)$",(-1,0),S); label("$\textbf{(D)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,1)--(1,0)--(0,-1)--(-1,0)--cycle); label("$(0,1)$",(0,1),NE); label("$(1,0)$",(1,0),SE); label("$(0,-1)$",(0,-1),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(E)}$",(-2,1.5)); [/asy]

A finite non-empty set of integers is called $3$-[i]good[/i] if the sum of its elements is divisible by $3$. Find the number of $3$-good subsets of $\{0,1,2,\ldots,9\}$.

For some integer $n > 0$, a square paper of side length $2^{n}$ is repeatedly folded in half, right-to-left then bottom-to-top, until a square of side length 1 is formed. A hole is then drilled into the square at a point $\tfrac{3}{16}$ from the top and left edges, and then the paper is completely unfolded. The holes in the unfolded paper form a rectangular array of unevenly spaced points; when connected with horizontal and vertical line segments, these points form a grid of squares and rectangles. Let $P$ be a point chosen randomly from \textit{inside} this grid. Suppose the largest $L$ such that, for all $n$, the probability that the four segments $P$ is bounded by form a square is at least $L$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers $n$ such that $n$ divides the number of the year in which the $n$-th Olympiad takes place.

[u]Round 6[/u] [b]p16.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. There exist two possible points $X$ on $CA$ such that if $Y$ and $Z$ are the feet of the perpendiculars from $X$ to $AB$ and $BC,$ respectively, then the area of triangle $XY Z$ is $1$. If the distance between those two possible points can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $gcd(a, c) = 1$, then find $a +b+ c$. [b]p17.[/b] Let $f(n)$ be the number of orderings of $1,2, ... ,n$ such that each number is as most twice the number preceding it. Find the number of integers $k$ between $1$ and $50$, inclusive, such that $f (k)$ is a perfect square. [b]p18.[/b] Suppose that $f$ is a function on the positive integers such that $f(p) = p$ for any prime p, and that $f (xy) = f(x) + f(y)$ for any positive integers $x$ and $y$. Define $g(n) = \sum_{k|n} f (k)$; that is, $g(n)$ is the sum of all $f(k)$ such that $k$ is a factor of $n$. For example, $g(6) = f(1) + 1(2) + f(3) + f(6)$. Find the sum of all composite $n$ between $50$ and $100$, inclusive, such that $g(n) = n$. [u]Round 7[/u] [b]p19.[/b] AJ is standing in the center of an equilateral triangle with vertices labelled $A$, $B$, and $C$. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record $21$ labels in total, including the initial one. Find the number of distinct possible ordered triples $(a, b, c)$, where a is the number of $A$'s they recorded, b is the number of $B$'s they recorded, and c is the number of $C$'s they recorded. [b]p20.[/b] Let $S = \sum_{n=1}^{\infty} (1- \{(2 + \sqrt3)^n\})$, where $\{x\} = x - \lfloor x\rfloor$ , the fractional part of $x$. If $S =\frac{\sqrt{a} -b}{c}$ for positive integers $a, b, c$ with $a $ squarefree, find $a + b + c$. [b]p21.[/b] Misaka likes coloring. For each square of a $1\times 8$ grid, she flips a fair coin and colors in the square if it lands on heads. Afterwards, Misaka places as many $1 \times 2$ dominos on the grid as possible such that both parts of each domino lie on uncolored squares and no dominos overlap. Given that the expected number of dominos that she places can be written as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$, find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 4-5 [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$.

Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.) [i]Proposed by Ankan Bhattacharya, USA[/i]

Fames is playing a computer game with falling two-dimensional blocks. The playing field is $7$ units wide and infinitely tall with a bottom border. Initially the entire field is empty. Each turn, the computer gives Fames a $1\times 3$ solid rectangular piece of three unit squares. Fames must decide whether to orient the piece horizontally or vertically and which column(s) the piece should occupy ($3$ consecutive columns for horizontal pieces, $1$ column for vertical pieces). Once he confirms his choice, the piece is dropped straight down into the playing field in the selected columns, stopping all three of the piece's squares as soon as the piece hits either the bottom of the playing field or any square from another piece. All of the pieces must be contained completely inside the playing field after dropping and cannot partially occupy columns. If at any time a row of $7$ spaces is all filled with squares, Fames scores a point. Unfortunately, Fames is playing in [i]invisible mode[/i], which prevents him from seeing the state of the playing field or how many points he has, and he has already arbitrarily dropped some number of pieces without remembering what he did with them or how many there were. For partial credit, find a strategy that will allow Fames to eventually earn at least one more point. For full credit, find a strategy for which Fames can correctly announce "I have earned at least one more point" and know that he is correct.

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

Unit square $ZINC$ is constructed in the interior of hexagon $CARBON$. What is the area of triangle $BIO$?

Prove that the four segments connecting the vertices of the tetrahedron with the centers of gravity of the opposite faces have a common point.

In triangle $ABC$, points $P$, $Q$, and $R$ are marked on the sides $AB$, $BC$, and $AC$ respectively. The lengths of the sides of triangle $PQR$ are known to be 7, 8, and 9 centimeters. Find the radii of the circles inscribed in triangles $APR$, $BPQ$, and $CQR$ given that all three circles are tangent to the incircle of triangle $PQR$. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.

A cylindrical oil tank, lying horizontally, has an interior length of $ 10$ feet and an interior diameter of $ 6$ feet. If the rectangular surface of the oil has an area of $ 40$ square feet, the depth of the oil is: $ \textbf{(A)}\ \sqrt{5} \qquad \textbf{(B)}\ 2\sqrt{5} \qquad \textbf{(C)}\ 3\minus{}\sqrt{5} \qquad \textbf{(D)}\ 3\plus{}\sqrt{5} \\ \textbf{(E)}\ \text{either }3\minus{}\sqrt{5}\text{ or }3\plus{}\sqrt{5}$

When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

Let $f(x)$ be a real-valued function satisfying $af(x)+bf(-x)=px^2+qx+r$. $a$ and $b$ are distinct real numbers and $p,q,r$ are non-zero real numbers. Then $f(x)=0$ will have real solutions when (A)$\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}$ (B)$\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}$ (C)$\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}$ (D)$\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}$

A pile of cards, numbered with $1$, $2$, ..., $n$, is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$, then we reverse the order of the $k$ uppermost cards. Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.

Let $ z_1,z_2,z_3\in\mathbb{C} , $ different two by two, having the same modulus $ \rho . $ Show that: $$ \frac{1}{\left| z_1-z_2\right|\cdot \left| z_1-z_3\right|} +\frac{1}{\left| z_2-z_1\right|\cdot \left| z_2-z_3\right|} +\frac{1}{\left| z_3-z_1\right|\cdot \left| z_3-z_2\right|}\ge\frac{1}{\rho^2} . $$

Consider a quartet of positive numbers $(a,b,c,d)$. In one step, we transform it to $(ab,bc,cd,da)$. Prove that you can never obtain the initial set if neither of $a,b,c,d$ is $1$.

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

Determine all functions $f: \mathbb{R} \to \mathbb{R}$, such that $$f(xf(y)+1)=y+f(f(x)f(y))$$ for all $x, y \in \mathbb{R}$. (Theresia Eisenkölbl)

Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$