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}$

Which of the following four statements are true and which are false? a) If a polygon inscribed in a circle is equilateral, then it is also equiangular. b) If a polygon inscribed in a circle is equiangular, then it is also equilateral. c) If a polygon circumscribed to a circle is equilateral, then it is also equiangular. d) If a polygon circumscribed to a circle is equiangular, then it is also equilateral.

Let $a_1\in(0,1)$ and define recursively the sequence $(a_n)_{n\geq 1}$ by $a_{n+1}=3a_n-4a_n^3$ for all $n\geq 1$. a) Prove that for all $n$ we have $|a_n|<1$. b) Prove that for any $k\geq 2$ we can choose $a_1\in(0,1)$ adequately such that $a_{n+k}=a_n$ for all $n\geq 1$. [i]Sergiu Moroianu[/i]

$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$ passing through $O$, where $X$ and $X'$ are boundary points of $A$, then \[ \frac12 \leq \frac {OX}{OX'} \leq 2.\]

Let $\alpha$ be the positive root of the equation $x^2+x=5$. Let $n$ be a positive integer number, and let $c_0,c_1,\ldots,c_n\in \mathbb{N}$ be such that $ c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015. $ a. Prove that $c_0+c_1+c_2+\cdots+c_n\equiv 2 \pmod{3}$. b. Find the minimum value of the sum $c_0+c_1+c_2+\cdots+c_n$.

Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$

In the triangle $ABC$, the cevians $AA_1$, $BB_1$ and $CC_1$ intersect at the point $O$. It turned out that $AA_1$ is the bisector, and the point $O$ is closer to the straight line $AB$ than to the straight lines $A_1C_1$ and $B_1A_1$. Prove that $\angle{BAC} > 120^{\circ}$.

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

$25.$ Let $C$ be the answer to Problem $27.$ What is the $C$-th smallest positive integer with exactly four positive factors? $26.$ Let $A$ be the answer to Problem $25.$ Determine the absolute value of the difference between the two positive integer roots of the quadratic equation $x^2-Ax+48=0$ $27.$ Let $B$ be the answer to Problem $26.$ Compute the smallest integer greater than $\frac{B}{\pi}$

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.