Toss the coin $n$ times, assume that each time, only appear only head or tail Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times $(1)$ Find $a(2016)$ and $b(2016)$ $(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$

$8$ CPLP football teams competed in a championship in which each team played one and only time with each of the other teams. In football, each win is worth $3$ points, each draw is worth $1$ point and the defeated team does not score. In that championship four teams were in first place with $15$ points and the others four came in second with $N$ points each. Knowing that there were $12$ draws throughout the championship, determine $N$.

Vanya was asked to write on the board an expression equal to $10$, using only the numbers $1$, the signs $+$ and $-$ and brackets (you cannot make up the numbers $11$, $111$, etc., as well as $(-1)$). He knows that the bully Anton will then correct all the $+$ signs to $-$ and vice versa. Help Vanya compose the required expression, which will remain equal to $10$ even after Anton's actions.

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]

There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.

In a horizontal strip $1 \times n$ made of $n$ unit squares the vertices of all squares are marked. The strip is partitioned into parts by segments connecting marked points and not lying on the sides of the strip. The segments can not have common inner points; the upper end of each segment must be either above the lower end or further to the right. Prove that the number of all partitions is divisible by $2^n$. (The partition where no segments are drawn, is counted too.) [i](E. Robeva, M. Sun)[/i]

A bag contains $12$ marbles: $3$ red, $4$ green, and $5$ blue. Repeatedly draw marbles with replacement until you draw two marbles of the same color in a row. What is the expected number of times that you will draw a marble?

An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the tunnels connecting them. Find the maximum number of good groups.

A polygon (not necessarily convex) on the coordinate plane is called [i]plump[/i] if it satisfies the following conditions: $\bullet$ coordinates of vertices are integers; $\bullet$ each side forms an angle of $0^\circ$, $90^\circ$, or $45^\circ$ with the abscissa axis; $\bullet$ internal angles belong to the interval $[90^\circ, 270^\circ]$. Prove that if a square of each side length of a plump polygon is even, then such a polygon can be cut into several convex plump polygons. [i](A. Yuran)[/i]

Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions. (i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$ (iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$

There are 100 persons in a hall, everyone knowing at least 67 of the others. Prove that there always exist four of them who know each other

The numbers $1,\, 2,\, \ldots\, , n^{2}$ are written in the squares of an $n \times n$ board in some order. Initially there is a token on the square labelled with $n^{2}.$ In each step, the token can be moved to any adjacent square (by side). At the beginning, the token is moved to the square labelled with the number $1$ along a path with the minimum number of steps. Then it is moved to the square labelled with $2,$ then to square $3,$ etc, always taking the shortest path, until it returns to the initial square. If the total trip takes $N$ steps, find the smallest and greatest possible values of $N.$

[u]Round 5[/u] [b]p13.[/b] Given that $x^3 + y^3 = 208$ and $x + y = 4$, what is the value of $\frac{1}{x} +\frac{1}{y}$? [b]p14.[/b] Find the sum of all three-digit integers $n$ such that the value of $n$ is equal to the sum of the factorials of $n$’s digits. [b]p15.[/b] Three christmas lights are initially off. The Grinch decides to fiddle around with the lights, switching one of the lights each second. He wishes to get every possible combination of lights. After how many seconds can the Grinch complete his task? [u]Round 6[/u] [b]p16.[/b] A regular tetrahedron of side length $1$ has four similar tetrahedrons of side length $1/2$ chopped off, one from each of the four vertices. What is the sum of the numbers of vertices, edges, and faces of the remaining solid? [b]p17.[/b] Mario serves a pie in the shape of a regular $2013$-gon. To make each slice, he must cut in a straight line starting from one vertex and ending at another vertex of the pie. Every vertex of a slice must be a vertex of the original $2013$-gon. If every person eats at least one slice of pie regardless of the size, what is the maximum number of people the $2013$-gon pie can feed? [b]p18.[/b] Find the largest integer $x$ such that $x^2 + 1$ divides $x^3 + x - 1000$. [u]Round 7[/u] [b]p19.[/b] In $\vartriangle ABC$, $\angle B = 87^o$, $\angle C = 29^o$, and $AC = 37$. The perpendicular bisector of $\overline{BC}$ meets $\overline{AC}$ at point $T$. What is the value of $AB + BT$? [b]p20.[/b] Consider the sequence $f(1) = 1$, $f(2) = \frac12$ ,$ f(3) =\frac{1+3}{2}$, $f(4) =\frac{ 1+3}{2+4}$ ,$ f(5) = \frac{ 1+3+5}{2+4} . . . $ What is the minimum value of $n$, with $n > 1$, such that $|f(n) - 1| \le \frac{1}{10 }$. [b]p21.[/b] Three unit circles are centered at $(0, 0)$,$(0, 2)$, and $(2, 0)$. A line is drawn passing through $(0, 1)$ such that the region inside the circles and above the line has the same area as the region inside the circles and below the line. What is the equation of this line in $y = mx + b$ form? [u]Round 8[/u] [b]p22.[/b] The two walls of a pinball machine are positioned at a $45$ degree angle to each other. A pinball, represented by a point, is fired at a wall (but not at the intersection of the two walls). What is the maximum number of times the ball can bounce off the walls? [b]p23.[/b] Albert is fooling people with his weighted coin at a carnival. He asks his guests to guess how many times heads will show up if he flips the coin $4$ times. Richard decides to play the game and guesses that heads will show up $2$ times. In the previous game, Zach guessed that the heads would show up 3 times. In Zach’s game, there were least 3 heads, and given this information, Zach had a $\frac49$ chance of winning. What is the probability that Richard guessescorrectly? [b]p24.[/b] Let $S$ be the set of all positive integers relatively prime to $2013$ that have no prime factor greater than $15$. Find the sum of the reciprocals of all of the elements of $S$. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?

Write two ones, then a $2$ between them, then a $3$ between the numbers whose sum is $3$, then a $4$ between the numbers whose sum is $4$, as shown below: $$(1, 1), (1, 2, 1),(1, 3, 2, 3, 1), (1, 4, 3, 2, 3, 4, 1)$$ and so on. Prove that the number of times $n$ appears, ($n\ge 2$), is equal to the number of positive integers less than $n$ and relative prime with $n$..

$100$ weights, measuring $1,2, ..., 100$ grams, respectively, are placed in the two pans of a scale such that the scale is balanced. Prove that two weights can be removed from each pan such that the equilibrium is not broken.

In a $ 4 \times 4 $ grid of sixteen unit squares, exactly $8$ are shaded so that each shaded square shares an edge with exactly one other shaded square. How many ways can this be done?

The set $ S$ consists of $ n > 2$ points in the plane. The set $ P$ consists of $ m$ lines in the plane such that every line in $ P$ is an axis of symmetry for $ S$. Prove that $ m\leq n$, and determine when equality holds.

[u]Part 1[/u] [b]p1.[/b] Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. What is the sum of all $n$ for which $B$ has a winning strategy? [b]p2.[/b] How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels? [b]p3.[/b] Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$. [b]p4.[/b] Lisa has a $3$D box that is $48$ units long, $140$ units high, and $126$ units wide. She shines a laser beam into the box through one of the corners, at a $45^o$ angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box. [u]Part 2[/u] [b]p5.[/b] How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon? [b]p6.[/b] Let $a$ be the greatest root of $y = x^3 + 7x^2 - 14x - 48$. Let $b$ be the number of ways to pick a group of $a$ people out of a collection of $a^2$ people. Find $\frac{b}{2}$ . [b]p7.[/b] Consider the equation $$1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$ with $a, b, c$, and $d$ being positive integers. What is the largest value for $d$? [b]p8.[/b] The number of non-negative integers $x_1, x_2,..., x_{12}$ such that $$x_1 + x_2 + ... + x_{12} \le 17$$ can be expressed in the form ${a \choose b}$ , where $2b \le a$. Find $a + b$. [u]Part 3[/u] [b]p9.[/b] In the diagram below, $AB$ is tangent to circle $O$. Given that $AC = 15$, $AB = 27/2$, and $BD = 243/34$, compute the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png[/img] [b]p10.[/b] If $$\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2, $$ where $\log x$ is the base-$10$ logarithm of $x$, then it follows that $x =\sqrt{n}$. Compute $n^2$. [b]p11.[/b] [b]p12.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5, $$ where $n$ is an integer less than $170$. [u]Part 4[/u] [b]p13.[/b] Let $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Define $f(n)$ as the number of distinct two-digit integers that can be formed from digits in $n$. For example, $f(15) = 4$ because the integers $11$, $15$, $51$, $55$ can be formed from digits of $15$. Let $w$ be such that $f(3xz - w) = w$. Find $w$. [b]p14.[/b] Let $w$ be the answer to number $13$ and $z$ be the answer to number $16$. Let $x$ be such that the coefficient of $a^xb^x$ in $(a + b)^{2x}$ is $5z^2 + 2w - 1$. Find $x$. [b]p15.[/b] Let $w$ be the answer to number $13$, $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Let $A$, $B$, $C$, $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. Now, let $AE = 3x$, $ED = w^2 - w + 1$, and $AD = 2z$. If $FG = y$, find $y$. [b]p16.[/b] Let $w$ be the answer to number $13$, and $x$ be the answer to number $16$. Let $z$ be the number of integers $n$ in the set $S = \{w,w + 1, ... ,16x - 1, 16x\}$ such that $n^2 + n^3$ is a perfect square. Find $z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$? [b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors? [b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$? [b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain? [b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$) [b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$. [b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides? [b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent? [b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$. [b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$? [b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$? [b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon? [b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important? [b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$. [b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/ [b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles? [b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ . [b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer? [b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals? [b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.

Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies $$2\mid x_1,x_2,\ldots,x_k$$ $$x_1+x_2+\ldots+x_k=n$$ then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.

For a group of 5 girls, denoted as $G_1,G_2,G_3,G_4,G_5$ and $12$ boys. There are $17$ chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met: (a) Each chair has a proper seat. (b) The order, from left to right, of the girls seating is $G_1; G_2; G_3; G_4; G_5.$ (c) Between $G_1$ and $G_2$ there are at least three boys. (d) Between $G_4$ and $G_5$ there are at least one boy and most four boys. How many such arrangements are possible?

There are ten cards with the number $a$ on each, ten with $b$ and ten with $c$, where $a, b$ and $c$ are distinct real numbers. For every five cards, it is possible to add another five cards so that the sum of the numbers on these ten cards is $0$. Prove that one of $a, b$ and $c$ is $0$.

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.