A complete graph of $n$ vertices is a set of $n$ vertices and those vertices are connected in pairs by edges. Suppose the graph has $n$ vertices $A_1, A_2, ..., A_n$, the cycle is a set of edges of the form $A_{i_1}A_{i_2}, A_{i_2}A_{i_3},..., A_{i_m}A_{i_1}$ with $i_1, i_2, ..., i_m \in {1, 2, ..., n}$ double one different. We call $m$ the length of this cycle. Find the smallest positive integer$ n$ such that for every way of coloring all edges of a complete graph of $n$ vertices, each edge filled with one of three different colors, there is always a cycle of even length with the same color. PS. The same problem with another wording [url=https://artofproblemsolving.com/community/c6h151391p852296]here [/url].

Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$.

A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree? [asy] filldraw(circle((0,0),1),gray(0.7)); draw((0,0)--(0.81915, 0.57358)); dot((0.81915, 0.57358)); draw((0.81915, 0.57358)--(0.475006, 1.06507)); arrow((0.417649,1.14698), dir(305), 12); draw((0,0)--(1,0),dashed); label("$\theta$", (0.2, 0.2/3), fontsize(8)); label("$r$", (0.409575,0.28679), NW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant rate, how many hours would it have taken Alex to rake the entire lawn by himself?

Evaluate the following definite integral. \[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]

Find the remainder when \[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\] is divided by $ 1000$.

A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.

Represent the hypotenuse of a right triangle by $ c$ and the area by $ A$. The atltidue on the hypotenuse is: $ \textbf{(A)}\ \frac{A}{c} \qquad \textbf{(B)}\ \frac{2A}{c} \qquad \textbf{(C)}\ \frac{A}{2c} \qquad \textbf{(D)}\ \frac{A^2}{c} \qquad \textbf{(E)}\ \frac{A}{c^2}$

Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.

Charles is playing a variant of Sudoku. To each lattice point $(x, y)$ where $1\le x,y <100$, he assigns an integer between $1$ and $100$ inclusive. These integers satisfy the property that in any row where $y=k$, the $99$ values are distinct and never equal to $k$; similarly for any column where $x=k$. Now, Charles randomly selects one of his lattice points with probability proportional to the integer value he assigned to it. Compute the expected value of $x+y$ for the chosen point $(x, y)$.

Given are the points $A$ and $B$ on the edge of a circular pool. The athlete has to get from point $A$ to point $B$ by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?

Find all solutions (real and complex) for $x,y,z$, given that: \[ x+y+z = 6 \\ x^2+y^2+z^2 = 12 \\ x^3+y^3+z^3 = 24 \]

Let $a$ be a positive integer.Suppose that $\forall n$ ,$\exists d$, $d\not =1$, $d\equiv 1\pmod n$ ,$d\mid n^2a-1$.Prove that $a$ is a perfect square.

What is the largest possible value of $\dfrac{x^2+2x+6}{x^2+x+5}$ where $x$ is a positive real number? $ \textbf{(A)}\ \dfrac{14}{11} \qquad\textbf{(B)}\ \dfrac{9}{7} \qquad\textbf{(C)}\ \dfrac{13}{10} \qquad\textbf{(D)}\ \dfrac{4}{3} \qquad\textbf{(E)}\ \text{None of the preceding} $

Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$. [i]Proposed by Andrew Wen[/i]

$ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$. $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$. Given $EF=2$ and $FD=1$. Find the area of the rhombus $ABCD$

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one. If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!! Show that, we can repeat this operation until we have all $1996$ numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!!

One ticket to a show costs $ \$20$ at full price. Susan buys 4 tickets using a coupon that gives her a $25\%$ discount. Pam buys 5 tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$

Let $ A_0\equal{}(0,0)$. Distinct points $ A_1,A_2,\ldots$ lie on the $ x$-axis, and distinct points $ B_1,B_2,\ldots$ lie on the graph of $ y\equal{}\sqrt{x}$. For every positive integer $ n$, $ A_{n\minus{}1}B_nA_n$ is an equilateral triangle. What is the least $ n$ for which the length $ A_0A_n\ge100$? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 21$

The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). How many $4$-digit numbers include $4$ consecutive digits? [i](9 and 0 are not considered consecutive digits.)[/i] $\text{(A) }18\qquad\text{(B) }24\qquad\text{(C) }144\qquad\text{(D) }162\qquad\text{(E) }168$

Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Let $a=BC, b=CA,c=AB.$ Prove that: ${a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .$

There are $20$ collinear points, separated by the same distance: $$. \,\,\, . \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\, .$$ Miguel has to paint three or more of these points red, in such a way that the red points are separated by the same distance and it is impossible to paint exactly one more point red without violating the previous condition. Determine in how many ways Miguel can do his homework.

Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? $\text{(A)}\ 30 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 70$

Suppose we have $w < x < y < z$, and each of the $6$ pairwise sums are distinct. The $4$ greatest sums are $4, 3, 2, 1$. What is the sum of all possible values of $w$?