Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $

Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are all integers.

If $ a$ and $ b$ are two unequal positive numbers, then: $ \textbf{(A)}\ \frac {2ab}{a \plus{} b} > \sqrt {ab} > \frac {a \plus{} b}{2} \qquad\textbf{(B)}\ \sqrt {ab} > \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2}$ $ \textbf{(C)}\ \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2} > \sqrt {ab} \qquad\textbf{(D)}\ \frac {a \plus{} b}{2} > \frac {2ab}{a \plus{} b} > \sqrt {ab}$ $ \textbf{(E)}\ \frac {a \plus{} b}{2} > \sqrt {ab} > \frac {2ab}{a \plus{} b}$

In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.

Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$, where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$. Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$. Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$, where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$.

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

Let $x, y, z$ be positive real numbers such that $\frac{1}{1 + x}+\frac{1}{1 + y}+\frac{1}{1 + z}= 2$. Prove that $8xyz \le 1$.

[b]a)[/b] Prove that $$ k+\frac{1}{2}-\frac{1}{8k}<\sqrt{k^2+k}<k+\frac{1}{2}-\frac{1}{8k}+\frac{1}{16k^2} , $$ for any natural number $ k. $ [b]b)[/b] Prove that there exists four numbers $ \alpha,\beta,\gamma,\delta\in\{0,1,2,3,4,5,6,7,8,9\} $ such that $$ \left\lfloor\sum_{k=1}^{2012} \sqrt{k(k+1)\left( k^2+k+1 \right)}\right\rfloor =\underbrace{\ldots\alpha \beta\gamma\delta}_{\text{decimal form}} $$ and $ \alpha +\delta =\gamma . $

The picture shows a triangle divided into $25$ smaller triangles, numbered $1$ to $25$. Is it possible to place the same numbers in the square cells 5$\times 5$ so that any two numbers written in adjacent triangles were are also written in adjacent cells of the square? (The cells of a square are considered adjacent if they have a common side.) [img]https://cdn.artofproblemsolving.com/attachments/4/3/758fe5531ab3e576ef4712c095b393f8dff397.png[/img]

Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere?

Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

Equilateral triangle $ABC$ has side length $20$. Let $PQRS$ be a square such that $A$ is the midpoint of $\overline{RS}$ and $Q$ is the midpoint of $\overline{BC}$. Compute the area of $PQRS$.

While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a $2020$-term strictly increasing geometric sequence with an integer common ratio . Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_0$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_0$, what is the closest integer to $\log_{2} d$?

Solve the given system in prime numbers \begin{align*} x^2+yu = (x+u)^v \end{align*} \begin{align*} x^2+yz=u^4 \end{align*}