Solve the equation $$arc \sin (\sin x) + arc \cos (\cos x)=0$$

Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$. Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$. [hide="Comment"]In the contest, the problem was given with a) and b): a) Prove the above for $n=2$; b) Prove the above for $n\geq 3$ as well.[/hide]

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

Triangles $ ABC$ and $ A_1B_1C_1$ have the same area. Using compass and ruler, can we always construct triangle $ A_2B_2C_2$ equal to triangle $ A_1B_1C_1$ so that the lines $ AA_2$, $ BB_2$, and $ CC_2$ are parallel?

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.

Judge the following statements. If true, prove it; If false, give a counter-example. [b](a)[/b] $P,Q$ are two different points on the sime side of line $l$. There exists two different circles, passing $P,Q$, and tangent to line $l$. [b](b)[/b] If $a>0,b>0,a\neq1,b\neq1$, then$\log_ab+\log_ba\geq2$. [b](c)[/b] $A,B$ are two sets of points on coordinate plane. $C_r=\{(x,y)|x^2+y^2\leq r^2\}$. For any $r\geq0$, $C_r\cup A\subset C_r\cup B$, then $A\subset B$.

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? [i]Proposed by Mihai Bălună, Romania[/i]

The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$

For four pairwise different positive integers $a,b,c$ and $d$ six numbers are calculated: $ab+10$,$ac+10$,$ad+10$,$bc+10$,$bd+10$ and $cd+10$. Find the maximum amount of them which can be perfect squares.

Dorothy organized a party for the birthday of Duck Mom and she also prepared a cylindershaped cake. Since she was originally expecting to have $15$ guests, she divided the top of the cake into this many equal circular sectors, marking where the cuts need to be made. Just for fun Dorothy’s brother Donald split the top of the cake into $10$ equal circular sectors in such a way that some of the radii that he marked coincided with Dorothy’s original markings. Just before the arrival of the guests Douglas cut the cake according to all markings, and then he placed the cake into the fridge. This way they forgot about the cake and only got to eating it when only $6$ of them remained. Is it possible for them to divide the cake into $6$ equal parts without making any further cuts?

$O$ is a circumcircle of $ABC$ and $CO$ meets $AB$ at $P$, and $BO$ meets $AC$ at $Q$. Show that $BP=PQ=QC$ if and only if $\angle A=60^{\circ}$.

Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge 3k/4$. Let $n$ be a composite integer. Prove: (a) if $n=2p_k$, then $n$ does not divide $(n-k)!$; (b) if $n>2p_k$, then $n$ divides $(n-k)!$.

The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that (i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$; (ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$; (iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$. Prove that $a_1,a_2$ and $a_3$ cannot all be prime.

Convex octagon $AC_1BA_1CB_1$ satisfies: $AB_1=AC_1$, $BC_1=BA_1$, $CA_1=CB_1$ and $\angle{A}+\angle{B}+\angle{C}=\angle{A_1}+\angle{B_1}+\angle{C_1}$. Prove that the area of $\triangle{ABC}$ is equal to half the area of the octagon.

Let $n \ge 3$ be an odd integer. Each cell is a $n \times n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ [i]path [/i] if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.

Given a point $ P$ inside a triangle $ \triangle ABC$. Let $ D$, $ E$, $ F$ be the orthogonal projections of the point $ P$ on the sides $ BC$, $ CA$, $ AB$, respectively. Let the orthogonal projections of the point $ A$ on the lines $ BP$ and $ CP$ be $ M$ and $ N$, respectively. Prove that the lines $ ME$, $ NF$, $ BC$ are concurrent. [i]Original formulation:[/i] Let $ ABC$ be any triangle and $ P$ any point in its interior. Let $ P_1, P_2$ be the feet of the perpendiculars from $ P$ to the two sides $ AC$ and $ BC.$ Draw $ AP$ and $ BP,$ and from $ C$ drop perpendiculars to $ AP$ and $ BP.$ Let $ Q_1$ and $ Q_2$ be the feet of these perpendiculars. Prove that the lines $ Q_1P_2,Q_2P_1,$ and $ AB$ are concurrent.

Let $f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx $, where $a_1 ,a_2 ,\ldots,a_n $ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \leq | \sin x |$ for all real $x,$ prove that $$|a_1 +2a_2 +\cdots +na_{n}|\leq 1.$$

In $\triangle ABC$, $A,B,C$ are arithmetic sequence, and $c-a$ is equal to height on side $BC$, then $\sin\frac{C-A}{2}=$________.

A base $9$ number [i]probably places[/i] if it has a $7$ as one of its digits. Find the number of base $9$ numbers less than or equal to $100$ in base $10$ that probably place.

Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$ Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]