In $\triangle ABC$ , prove that\[\frac{tan\frac{A}{2}+tan\frac{B}{2}+tan\frac{C}{2}}{\sqrt{3}}\geq\sqrt[6]{tan^2\frac{A}{2}+tan^2\frac{B}{2}+tan^2\frac{C}{2}}.\]

A function $ f$ from $ [0,1]$ to $ \mathbb{R}$ has the following properties: $ (i)$ $ f(1)\equal{}1;$ $ (ii)$ $ f(x) \ge 0$ for all $ x \in [0,1]$; $ (iii)$ If $ x,y,x\plus{}y \in [0,1]$, then $ f(x\plus{}y) \ge f(x)\plus{}f(y)$. Prove that $ f(x) \le 2x$ for all $ x \in [0,1]$.

Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.

Emily draws six dots on a piece of paper such that no three lie on a straight line, then draws a line segment connecting each pair of dots. She then colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?

Let $n \geq 2$ be an integer. Consider an $n\times n$ chessboard with the usual chessboard colouring. A move consists of choosing a $1\times 1$ square and switching the colour of all squares in its row and column (including the chosen square itself). For which $n$ is it possible to get a monochrome chessboard after a finite sequence of moves?

If $ (0.2)^x \equal{} 2$ and $ \log 2 \equal{} 0.3010$, then the value of $ x$ to the nearest tenth is: $ \textbf{(A)}\ \minus{} 10.0 \qquad\textbf{(B)}\ \minus{} 0.5 \qquad\textbf{(C)}\ \minus{} 0.4 \qquad\textbf{(D)}\ \minus{} 0.2 \qquad\textbf{(E)}\ 10.0$

Players $ A,B$ alternately move a knight on a $ 1994\times 1994$ chessboard. Player $ A$ makes only horizontal moves, i.e. such that the knight is moved to a neighboring row, while $ B$ makes only vertical moves. Initally player $ A$ places the knight to an arbitrary square and makes the first move. The knight cannot be moved to a square that was already visited during the game. A player who cannot make a move loses. Prove that player $ A$ has a winning strategy.

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}.$ Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$? $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{2}{5} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{3}{4} $

Let $n$ be an even integer greater than $2$. On the vertices of a regular polygon with n sides we can place red or blue chips. Two players, $A$ and $B$, play alternating turns of the next mode: each player, on their turn, chooses two vertices that have no tiles and places on one of them a red chip and in the other a blue chip. The goal of $A$ is to get three vertices consecutive with tiles of the same color. $B$'s goal is to prevent this from happening. To the beginning of the game there are no tiles in any of the vertices. Show that regardless of who starts to play, Player $B$ can always achieve his goal.

Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$ (Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)

Let $ABC$ be an isosceles triangle ($AB=BC$) and $\ell$ be a ray from $B$. Points $P$ and $Q$ of $\ell$ lie inside the triangle in such a way that $\angle BAP=\angle QCA$. Prove that $\angle PAQ=\angle PCQ$.

Determine all solutions of the linear system of equations \begin{align*} &x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\ -&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\ -&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\ &&&&&&&&&&&\vdots \\ -&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na, \end{align*} with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$

Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle EAB$? [asy] size(170); defaultpen(linewidth(0.6)); real r = 25; draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle); draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle); label("$A$",dir(135),NW); label("$B$",dir(45),NE); label("$C$",dir(315),SE); label("$D$",dir(225),SW); label("$E$",dir(135-r),N); label("$F$",dir(45-r),E); label("$G$",dir(315-r),S); label("$H$",dir(225-r),W); [/asy] $\textbf{(A) }20^\circ\qquad\textbf{(B) }30^\circ\qquad\textbf{(C) }32^\circ\qquad\textbf{(D) }35^\circ\qquad\textbf{(E) }45^\circ$

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (R Zhenodarov)

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

A block of mass $\text{m = 3.0 kg}$ slides down one ramp, and then up a second ramp. The coefficient of kinetic friction between the block and each ramp is $\mu_\text{k} = \text{0.40}$. The block begins at a height $\text{h}_\text{1} = \text{1.0 m}$ above the horizontal. Both ramps are at a $\text{30}^{\circ}$ incline above the horizontal. To what height above the horizontal does the block rise on the second ramp? (A) $\text{0.18 m}$ (B) $\text{0.52 m}$ (C) $\text{0.59 m}$ (D) $\text{0.69 m}$ (E) $\text{0.71 m}$

We are given $n$ points in space. Some pairs of these points are connected by line segments so that the number of segments equals $[n^2/4],$ and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle.

[b]p1[/b]. Consider the following four statements referring to themselves: 1. At least one of these statements is true. 2. At least two of these statements are false. 3. At least three of these statements are true. 4. All four of these statements are false. Determine which statements are true and which are false. Justify your answer. [b]p2.[/b] Let $f(x) = a_{2017}x^{2017} + a_{2016}x^{2016} + ... + a_1x + a_0$ where the coefficients $a_0, a_1, ... , a_{2017}$ have not yet been determined. Alice and Bob play the following game: $\bullet$ Alice and Bob alternate choosing nonzero integer values for the coefficients, with Alice going first. (For example, Alice’s first move could be to set $a_{18}$ to $-3$.) $\bullet$ After all of the coefficients have been chosen: - If f(x) has an integer root then Alice wins. - If f(x) does not have an integer root then Bob wins. Determine which player has a winning strategy and what the strategy is. Make sure to justify your answer. [b]p3.[/b] Suppose that a circle can be inscribed in a polygon $P$ with $2017$ equal sides. Prove that $P$ is a regular polygon; that is, all angles of $P$ are also equal. [b]p4.[/b] A $3 \times 3 \times 3$ cube of cheese is sliced into twenty-seven $ 1 \times 1 \times 1$ blocks. A mouse starts anywhere on the outside and eats one of the $1\times 1\times 1$ cubes. He then moves to an adjacent cube (in any direction), eats that cube, and continues until he has eaten all $27$ cubes. (Two cubes are considered adjacent if they share a face.) Prove that no matter what strategy the mouse uses, he cannot eat the middle cube last. [Note: One should neglect gravity – intermediate configurations don’t collapse.] p5. Suppose that a constant $c > 0$ and an infinite sequence of real numbers $x_0, x_1, x_2, ...$ satisfy $x_{k+1} =\frac{x_k + 1}{1 - cx_k}$ for all $k \ge 0$. Prove that the sequence $x_0, x_1, x_2, ....$ contains infinitely many positive terms and also contains infinitely many negative terms. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.

Let $n$ and $k$ be positive integers. An $n$-tuple $(a_1, a_2,\ldots , a_n)$ is called a permutation if every number from the set $\{1, 2, . . . , n\}$ occurs in it exactly once. For a permutation $(p_1, p_2, . . . , p_n)$, we define its $k$-mutation to be the $n$-tuple $$(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}),$$ where indices are taken modulo $n$. Find all pairs $(n, k)$ such that every two distinct permutations have distinct $k$-mutations. [i]Remark[/i]: For example, when $(n, k) = (4, 2)$, the $2$-mutation of $(1, 2, 4, 3)$ is $(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)$. [i]Proposed by Borna Šimić[/i]

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: [list] [*] Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged. [/list] Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user. [i]Proposed by Adrian Beker, Croatia[/i]

Find all real numbers $q$, such that for all real $p \geq 0$, the equation $x^2-2px+q^2+q-2=0$ has at least one real root in $(-1;0)$.

Given a $2\times 5$ rectangle is divided into unit squares as figure below. [img]https://cdn.artofproblemsolving.com/attachments/6/a/9432bbf40f6d89ee1cbb507e1a3f65326c6a13.png[/img] How many ways are there to write the letters $H, A, N, O, I$ into all of the unit squares, such that two neighbor squares (the squares with a common side) do not contain the same letters? (Each unit square is filled by only one letter and each letter may be used several times or not used as well.)

In triangle $ABC$ the midpoints of sides $AC, BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A, B$ and the orthocenter of triangle $ABC$.