Given the sequence $(u_n)$ satisfying:$$\left\{ \begin{array}{l} 1 \le {u_1} \le 3\\ {u_{n + 1}} = 4 - \dfrac{{2({u_n} + 1)}}{{{2^{{u_n}}}}},\forall n \in \mathbb{Z^+}. \end{array} \right.$$ Prove that: $1\le u_n\le 3,\forall n\in \mathbb{Z^+}$ and find the limit of $(u_n).$

A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square. [img]https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png[/img]

Given a prime $p$ and positive integer $k$, an integer $n$ with $0 \le n < p$ is called a $(p, k)$-Hofstadterian residue if there exists an infinite sequence of integers $n_0, n_1, n_2, \ldots$ such that $n_0 \equiv n$ and $n_{i + 1}^k \equiv n_i \pmod{p}$ for all integers $i \ge 0$. If $f(p, k)$ is the number of $(p, k)$-Hofstadterian residues, then compute $\displaystyle \sum_{k = 1}^{2016} f(2017, k)$. [i]Proposed by Ashwin Sah[/i]

A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.

Given a triangle $ABC$ with circumcircle $\Gamma$, and two arbitrary points $X, Y$ on $\Gamma$. Let $D$, $E$, $F$ be points on lines $BC$, $CA$, $AB$, respectively, such that $AD$, $BE$, and $CF$ concur at a point $P$. Let $U$ be a point on line $BC$ such that $X$, $Y$, $D$, $U$ are concyclic. Similarly, let $V$ be a point on line $CA$ such that $X$, $Y$, $E$, $V$ are concyclic, and let $W$ be a point on line $AB$ such that $X$, $Y$, $F$, $W$ are concyclic. Prove that $AU$, $BV$, $CW$ concur at a single point. [i]Proposed by chengbilly[/i]

Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$, $B_T$ and $C_T$ as the points where lines $AT$, $BT$, and $CT$ meet the circumcircle of $ABC$. Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$.

The SET game is a deck with $81$ unique cards that vary in four features across three possibilities for each kind of feature: shape (oval, squiggle or diamond), color (red, green or purple), number of shapes (one, two or three) and shading (solid, striped or open). A $\textbf{set}$ consists in three cards whose characteristics, when considered individually, are the same on each card or different on all of them. All features have to satisfy this rule. In other words: the shape must be the same on all three cards or different on all them, the color must be the same on the three cards or different on all them, and so on. Ana and Bárbara divided among themselves the $81$ SET cards. Ana got $40$ cards and Bárbara got $41$. Each girl counted the number of ways she could form a three-card $\textbf{set}$ with her cards. What are the possible values of the sum of these two numbers?

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

Solve the system of equations $$ \qquad<br /> \begin{array}{c}<br /> x_1x_2 = 1\\<br /> x_2x_3 = 2\\<br /> x_3x_4 = 3\\<br /> \ldots\\<br /> x_nx_1 = n<br /> \end{array}$$

Given are real numbers $a_1, a_2, \ldots, a_n$ ($n>3$), such that $a_k^3=a_{k+1}^2+a_{k+2}^2+a_{k+3}^2$ for all $k=1,2,...,n$. Prove that all numbers are equal.

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

a) An altitude of a triangle is also a tangent to its circumcircle. Prove that some angle of the triangle is larger than $90^o$ but smaller than $135^o$. b) Some two altitudes of the triangle are both tangents to its circumcircle. Find the angles of the triangle.

We say that a positive integer is an [i]almost square[/i], if it is equal to the product of two consecutive positive integers. Prove that every almost square can be expressed as a quotient of two almost squares. V. Senderov

In the Cartesian plane, let $S_{i,j} = \{(x,y)\mid i \le x \le j\}$. For $i=0,1,\ldots,2012$, color $S_{i,i+1}$ pink if $i$ is even and gray if $i$ is odd. For a convex polygon $P$ in the plane, let $d(P)$ denote its pink density, i.e. the fraction of its total area that is pink. Call a polygon $P$ [i]pinxtreme[/i] if it lies completely in the region $S_{0,2013}$ and has at least one vertex on each of the lines $x=0$ and $x=2013$. Given that the minimum value of $d(P)$ over all non-degenerate convex pinxtreme polygons $P$ in the plane can be expressed in the form $\frac{(1+\sqrt{p})^2}{q^2}$ for positive integers $p,q$, find $p+q$. [i]Victor Wang.[/i]

[u]Round 1[/u] [b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$? [b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated? [img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img] [b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time. [img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img] [b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img] [b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$? [u]Round 2[/u] [b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation. [img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img] Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one? [img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img] [b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

It's given system of equations $a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$ $a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$ .......... $a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$ such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it

In $\triangle ABC$, the interior sides of which are mirrors, a laser is placed at point $A_1$ on side $BC$. A laser beam exits the point $A_1$, hits side $AC$ at point $B_1$, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side $AB$ at point $C_1$, then side $BC$ at point $A_2$, then side $AC$ again at point $B_2$, then side $AB$ again at point $C_2$, then side $BC$ again at point $A_3$, and finally, side $AC$ again at point $B_3$. (a) Prove that $\angle B_3A_3C = \angle B_1A_1C$. (b) Prove that such a laser exists if and only if all the angles in $\triangle ABC$ are less than $90^{\circ}$.

Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

It is known that the bisector of the angle $\angle ADC$ of the inscribed quadrilateral $ABCD$ passes through the center of the circle inscribed in the triangle $ABC$. Let $M$ be an arbitrary point of the arc $ABC$ of the circle circumscribed around $ABCD$. Denote by $P$ and $Q$ the centers of the circles inscribed in the triangles $ABM$ and $BCM$. Prove that all triangles $DPQ$ obtained by moving point $M$ are similar to each other. Find the angle $\angle PDQ$ and ratio $BP : PQ$ if $\angle BAC = \alpha$, $\angle BCA = \beta$

Given a positive integer $n$, let $p(n)$ be the product of the non-zero digits of $n$. (If $n$ has only one digits, then $p(n)$ is equal to that digit.) Let \[ S=p(1)+p(2)+p(3)+\cdots+p(999). \] What is the largest prime factor of $S$?

There are $n > 2022$ cities in the country. Some pairs of cities are connected with straight two-ways airlines. Call the set of the cities {\it unlucky}, if it is impossible to color the airlines between them in two colors without monochromatic triangle (i.e. three cities $A$, $B$, $C$ with the airlines $AB$, $AC$ and $BC$ of the same color). The set containing all the cities is unlucky. Is there always an unlucky set containing exactly 2022 cities?

For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.

The equation $ \sqrt {x \plus{} 10} \minus{} \frac {6}{\sqrt {x \plus{} 10}} \equal{} 5$ has: $ \textbf{(A)}\ \text{an extraneous root between } \minus{} 5\text{ and } \minus{} 1 \\ \textbf{(B)}\ \text{an extraneous root between } \minus{} 10\text{ and } \minus{} 6 \\ \textbf{(C)}\ \text{a true root between }20\text{ and }25 \qquad\textbf{(D)}\ \text{two true roots} \\ \textbf{(E)}\ \text{two extraneous roots}$