Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that \[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]

In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and losing team gets $0$ points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows: \[x_1 \ge x_2 \ge \cdots \ge x_N . \] Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

The cricles $\Gamma_1$ and $\Gamma_2$ intersect in $ M $ and $N.$ A line that goes through $ M $ intersects the cricles $\Gamma_1$ and $\Gamma_2$ in $ A$ and $B$, such that $M\in(AB)$. The bisector of angle $ AMN $ intersects the circle $\Gamma_1$ in $D,$ and the bisector of angle $BMN$ intersects the circle $\Gamma_2$ in $C.$ Prove that the circle with diameter $CD$ splits the segment $AB$ in half.

Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\). [i]Proposed by Raymond Feng[/i]

[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$? [b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$. [b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$. [b]p4.[/b] Circles of radius $7$, $7$, $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices? [b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square. [b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. [b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. [b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers. [b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

A container is filled with a total of $51$ red and white balls and has at least $1$ red ball and $1$ white ball. The probability of picking up $3$ red balls and $1$ white ball, without replacement, is equivalent to the probability of picking up $1$ red ball and $2$ white balls, without replacement. Compute the original number of red balls in the container.

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

A positive natural number $n$ is said to be [i]comico[/i] if its prime factorization is $n = p_1p_2...p_k$, with $k\ge 3$, and also the primes $p_1,..., p_k$ they fulfill that $p_1 + p_2 = c^2_1$ $p_1 + p_2 + p_3 = c^2_2$ $...$ $p_1 + p_2 + ...+ p_n = c^2_{n-1}$ where $c_1, c_2, ..., c_{n-1}$ are positive integers where $c_1$ is not divisible by $7$. Find all comico numbers less than $10,000$.

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} +
\cdots
+ a^1 + 1 \mid a^{n!} + a^{(n-1)!} +
\cdots

+ a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$. Proposed by Ephram Chun

Let $ABC$ be a triangle, and let $D$ and $D_1$ be points on segment $BC$ such that $BD = CD_1$. Construct point $E$ such that $EC\perp BC$ and $ED\perp AC$. Similarly, construct point $F$ such that $FB\perp BC$ and $FD\perp AB$. Prove that $EF\perp AD_1$.

$n$ numbers are written down along the circumference. Their sum equals to zero, and one of them equals $1$. a) Prove that there are two neighbours with their difference not less than $n/4$. b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on $8/(n^2)$. c) Try to improve the previous estimation, i.e what number can be used instead of $8$? d) Prove that for $n=30$ there is a number that differs from the arithmetic mean of its two neighbours not less than on $2/113$, give an example of such $30$ numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on $2/113$.

Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.

Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning? $\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 45$

Prove that in each year , the $13^{th}$ day of some month occurs on a Friday .

Consider a parallelogram $[ABCD]$ such that $\angle DAB$ is an acute angle. Let $G$ be a point in line $AB$ different from $B$ such that $\overline{BC}=\overline{GC}$, and let $H$ be a point in line $BC$ different from $B$ such that $\overline{AB}=\overline{AH}$. Prove that triangle $[GDH]$ is isosceles.

The monetary unit of a certain country is called Reo, and all the coins circulating are integers values of Reos. In a group of three people, each one has 60 Reos in coins (but we don't know what kind of coins each one has). Each of the three people can pay each other any integer value between 1 and 15 Reos, including, perhaps with change. Show that the three persons together can pay exactly (without change) any integer value between 45 and 135 Reos, inclusive.

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

Which positive integers satisfy the following three conditions? a) The number consists of at least two digits. b) The last digit is not zero. c) Inserting a zero between the last two digits yields a number divisible by the original number.

Triangle $ABC$ has $\angle{A}=90^\circ$, $\angle{C}=30^\circ$, and $AC=12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle{CAD}=60^\circ$. Define points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively. Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$ for positive integers $a,b,c,d,$ where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal. Find $a+b+c+d$.

Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?