Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and \[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \] Denote \[ A_n=\frac{3}{3 \cdot a_n^2-1}. \] Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.

Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$, for $i\in \{2,\ldots,k\}$, where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$. Prove that the sums \[\sum_{i=1}^k a_in_i\] with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$.

Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

Suppose $a_1 =\frac16$ and $a_n = a_{n-1} - \frac{1}{n}+ \frac{2}{n + 1} - \frac{1}{n + 2}$ for $n > 1$. Find $a_{100}$.

Let $A$, $X$ and $Y$ be three non-collinear points on the plane. Construct with a straightedge and compass a square $ABCD$ such that $X$ is on the line $BC$ and $Y$ is on the line $CD$.

Let $ABC$ be a triangle such that $AB \neq AC$. The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$, respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$, respectively. Further, let $I$ be the incentre of the triangle $ABC$. Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$. [i]Radu Gologan[/i]

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy]

We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

Prove that for every $ n \in \mathbb{N}^*$ exists a multiple of $ n$, having sum of digits equal to $ n$.

Let $a,b,c\in \mathbb{R}^+$, prove that: $$\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}$$

Calculate the following indefinite integrals. [1] $\int \frac{x}{\sqrt{5-x}}dx$ [2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$ [3] $\int (\sin x+\cos x)^2dx$ [4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$ [5]$\int (\sin x+\sin 2x)^2 dx$

In the playboard shown beside, players $A$ and $B$ alternately fill the empty cells by integers, player $A$ starting. In each step the empty cell and the integer can be chosen arbitrarily. Show that player $A$ can always achieve that all the equalities hold after the last step. [img]https://cdn.artofproblemsolving.com/attachments/c/0/524195b1a8ab8457b72005a162f8124c2b1bd2.png[/img]

Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?

An [i]augmentation[/i] on a graph $G$ is defined as doing the following: - Take some set $D$ of vertices in $G$, and duplicate each vertex $v_i \in D$ to create a new vertex $v_i'$. - If there's an edge between a pair of vertices $v_i, v_j \in D$, create an edge between vertices $v_i'$ and $v_j'$. If there's an edge between a pair of vertices $v_i \in D$, $v_j \notin D$, you can choose to create an edge between $v_i'$ and $v_j$ but do not have to. A graph is called [i]reachable[/i] from $G$ if it can be created through some sequence of augmentations on $G$. Some graph $H$ has $n$ vertices and satisfies that both $H$ and the complement of $H$ are reachable from a complete graph of $2021$ vertices. If the maximum and minimum values of $n$ are $M$ and $m$, find $M+m$. [i]Proposed by Oliver Hayman[/i]

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

Find all ordered pairs $(x,y)$ such that \[(x-2y)^2 + (y-1)^2 = 0. \]

For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$ $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$

Let $ABC$ be a triangle with $\angle BAC = 90^{\circ}$ and $\angle ACB = 54^{\circ}.$ We construct bisector $BD (D \in AC)$ of angle $ABC$ and consider point $E \in (BD)$ such that $DE = DC.$ Show that $BE = 2 \cdot AD.$

Let $A$ be a set of positive integers satisfying the following properties: (i) if $m$ and $n$ belong to $A$, then $m+n$ belong to $A$; (ii) there is no prime number that divides all elements of $A$. (a) Suppose $n_1$ and $n_2$ are two integers belonging to $A$ such that $n_2-n_1 >1$. Show that you can find two integers $m_1$ and $m_2$ in $A$ such that $0< m_2-m_1 < n_2-n_1$ (b) Hence show that there are two consecutive integers belonging to $A$. (c) Let $n_0$ and $n_0+1$ be two consecutive integers belonging to $A$. Show that if $n\geq n_0^2$ then $n$ belongs to $A$.

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$ [i]Proposed by Dusan Djukic $IMO \ Shortlist \ 2013$[/i]

For $n \ge 3$, it is given an $2n \times 2n$ board with black and white squares. It is known that all border squares are black and no $2 \times 2$ subboard has all four squares of the same color. Prove that there exists a $2 \times 2$ subboard painted like a chessboard, i.e. with two opposite black corners and two opposite white corners.

Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]

Consider the sum \[ S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, . \] Determine $\lfloor S_{4901} \rfloor$. Recall that if $x$ is a real number, then $\lfloor x \rfloor$ (the [i]floor[/i] of $x$) is the greatest integer that is less than or equal to $x$.