Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

By adding the same constant to $20,50,100$ a geometric progression results. The common ratio is: $ \textbf{(A)}\ \frac53 \qquad\textbf{(B)}\ \frac43\qquad\textbf{(C)}\ \frac32\qquad\textbf{(D)}\ \frac12\qquad\textbf{(E)}\ \frac13 $

Prove that $$AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R $ are any five points in a plane.

Show that a triangle whose side lengths are prime numbers cannot have integer area.

Let $JHIZ$ be a rectangle, and let $A$ and $C$ be points on sides $ZI$ and $ZJ,$ respectively. The perpendicular from $A$ to $CH$ intersects line $HI$ in $X$ and the perpendicular from $C$ to $AH$ intersects line $HJ$ in $Y.$ Prove that $X,$ $Y,$ and $Z$ are collinear (lie on the same line).

Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying [list] [*] $f(1) = 1$, [*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$. [/list] Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.

Three rods of lengths $1396, 1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?

Points $W$, $X$, $Y,$ and $Z$ are chosen inside a regular octagon so that four congruent rhombuses are formed, as shown in the diagram below. If the side length of the octagon is $1$, compute the area of quadrilateral $WXY Z$. [img]https://cdn.artofproblemsolving.com/attachments/9/6/bb12385cbd9fd802b3f3960b5e449268be45d4.png[/img]

The notation $\lfloor n \rfloor$ denotes the greatest integer less than or equal to $n$. Evaluate $\lfloor 2.1 \lfloor {-}4.3 \rfloor \rfloor$. $\textbf{(A) }{-}11\qquad\textbf{(B) }{-}10\qquad\textbf{(C) }{-}9\qquad\textbf{(D) }{-}8\qquad\textbf{(E) }{-}4$

The first $9$ positive integers are placed into the squares of a $3\times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest number. How many ways can we distribute the numbers into the chessboard such that $a=b=4$?

Let $\vartriangle A_1B_1C_1$ and $l_1, m_1, n_1$ be the trisectors closest to $A_1B_1$, $B_1C_1$, $C_1A_1$ of the angles $A_1, B_1, C_1$ respectively. Let $A_2 = l_1 \cap n_1$, $B_2 = m_1 \cap l_1$, $C_2 = n_1 \cap m_1$. So on we create triangles $\vartriangle A_nB_nC_n$ . If $\vartriangle A_1B_1C_1$ is equilateral prove that exists $n \in N$, such that all the sides of $\vartriangle A_nB_nC_n$ are parallel to the sides of $\vartriangle A_1B_1C_1$.

A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Justin is being served two different types of chips, A-chips, and B-chips. If there are $3$ B-chips and $5$ A-chips, and if Justin randomly grabs $3$ chips, what is the probability that none of them are A-chips?

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors. [i]Aleksandar Ivanov[/i]

If you walk for $ 45$ minutes at a rate of $ 4$ mph and then run for $ 30$ minutes at a rate of $ 10$ mph, how many miles have you gone at the end of one hour and $ 15$ minutes? \[ \textbf{(A)}\ 3.5 \text{ miles} \qquad \textbf{(B)}\ 8 \text{ miles} \qquad \textbf{(C)}\ 9 \text{ miles} \qquad \textbf{(D)}\ 25 \frac{1}{3} \text{ miles} \qquad \textbf{(E)}\ 480 \text{ miles} \]

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ . (b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.

Let $a$ and $b$ satisfy the conditions $\begin{cases} a^3 - 6a^2 + 15a = 9 \\ b^3 - 3b^2 + 6b = -1 \end{cases}$ . The value of $(a - b)^{2014}$ is: (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Let $n$ be a positive integer. We are given a $2n \times 2n$ table. Each cell is coloured with one of $2n^2$ colours such that each colour is used exactly twice. Jana stands in one of the cells. There is a chocolate bar lying in one of the other cells. Jana wishes to reach the cell with the chocolate bar. At each step, she can only move in one of the following two ways. Either she walks to an adjacent cell or she teleports to the other cell with the same colour as her current cell. (Jana can move to an adjacent cell of the same colour by either walking or teleporting.) Determine whether Jana can fulfill her wish, regardless of the initial configuration, if she has to alternate between the two ways of moving and has to start with a teleportation.

There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.

Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. [i]Proposed by Ivan Chan Kai Chin[/i]

In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle. [i]Proposed by Isabella Grabski [/i]

Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.