Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median.

Billy the kid likes to play on escalators! Moving at a constant speed, he manages to climb up one escalator in $24$ seconds and climb back down the same escalator in $40$ seconds. If at any given moment the escalator contains $48$ steps, how many steps can Billy climb in one second?

$\triangle{ABC}$ is equailateral. $E$ is any point on $\overline{AC}$ produced and the equilateral $\triangle{ECD}$ is drawn. If $M$ and $N$ are the midpoints of $\overline{AD}$ and $\overline{EB}$ respectively then show that $\triangle{CMN}$ is equilateral.

What is the sum of the leading (first) digits of the integers from $ 1$ to $2019$ when the integers are written in base $3$? Give your answer in base $10$.

Prove that if $f$ is a non-constant real-valued function such that for all real $x$, $f(x+1) + f(x-1) = \sqrt{3} f(x)$, then $f$ is periodic. What is the smallest $p$, $p > 0$ such that $f(x+p) = f(x)$ for all $x$?

There are $n$ hoops on a circle. Rik numbers all hoops with a natural number so that all numbers from $1$ to $n$ occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop $1$ and then follows the following rule: if he gets to hoop $k$, then he walks to the hoop that places $k$ clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way. For example, for $n = 6$ Rik can take a walk of length $5$ as the hoops are numbered as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/2/a/3d4b7edbba4d145c7e00368f9b794f39572dc5.png[/img] (a) Determine for every even $n$ how Rik can number the hoops so that he has one walk of length $n$. (b) Determine for every odd $n$ how Rik can number the hoops so that he has one walk of length $n - 1$. (c) Show that for an odd $n$ there is no such numbering of the hoops that Rik can make a walk of length $n$.

In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is $41$ years. After a $2010$ year old building was destroyed by lightning, the average age drops to $40$ years. How many trees were originally in the group? At most, how many of them were exactly $2010$ years old? (W. Janous, WRG Ursulinen, Innsbruck)

Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that: [list] [*] every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color; [*] every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color; [*] every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color; [*] every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color; [*] no two neighboring vertices have been colored in $1.$ and $3.$ color; [*] no two neighboring vertices have been colored in $2.$ and $4.$ color. [/list] Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$. Find $p+q$.

Let $ABC$ be a right triangle, with the right angle at $A$. The altitude from $A$ meets $BC$ at $H$ and $M$ is the midpoint of the hypotenuse $[BC]$. On the legs, in the exterior of the triangle, equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, prove that the angles $\angle NHP$ and $\angle AHQ$ are equal. Miguel Ochoa Sanchez and Leonard Giugiuc

Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded. [b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that $$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$ for any natural number $ n. $ Prove that the following relations are equivalent: $\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $ $ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $ [i]Eugen Păltănea[/i]

Let $ a,b,c$ be positive reals; show that $ \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right)$ [i](Darij Grinberg)[/i]

Suppose that $\{a_n\}$ is a sequence such that $a_{n+1}=(1+\frac{k}{n})a_{n}+1$ with $a_{1}=1$.Find all positive integers $k$ such that any $a_n$ be integer.

Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$. [i]Demetres Christofides, Cyprus[/i]

In a triangle $ABC$, the line symmetric to the median through $A$ with respect to the bisector of the angle at $A$ intersects $BC$ at $M$. Points $P$ on $AB$ and $Q$ on $AC$ are chosen such that $MP\parallel AC$ and $MQ\parallel AB$. Prove that the circumcircle of the triangle $MPQ$ is tangent to the line $BC$.

Asli will distribute $100$ candies among her brother and $18$ friends of him. Asli splits friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes same number of candies and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most how many candies can Asli's brother have? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18 $

Let $H$ denote the set of the matrices from $\mathcal{M}_n(\mathbb{N})$ and let $P$ the set of matrices from $H$ for which the sum of the entries from any row or any column is equal to $1$. a)If $A\in P$, prove that $\det A=\pm 1$. b)If $A_1,A_2,\ldots,A_p\in H$ and $A_1A_2\cdot \ldots\cdot A_p\in P$, prove that $A_1,A_2,\ldots,A_p\in P$.

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]. [i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.

Statement $P$: solution set to inequalities $a_1x^2+b_1x+c_1>0$ and $a_2x^2+b_2x+c_2>0$ are the same; statement $Q$: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$. $\text{(A)}$ $Q$ is sufficient and necessary condition of $P$. $\text{(B)}$ $Q$ is sufficient but unnecessary condition of $P$. $\text{(C)}$ $Q$ is insufficient but necessary condition of $P$. $\text{(D)}$ $Q$ is insufficient and unnecessary condition of $P$.

On the same side of a straight line three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is: $ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12 $

Let $P$ be a point on the interior of triangle $ABC$, such that the triangle $ABP$ satisfies $AP = BP$. On each of the other sides of $ABC$, build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $$BQ = QC$$ and $$CR = RA.$$ Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.

Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.