Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]

We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between $1 km/h$, $2km/h$,..., and $n km/h$, each one with a distinct velocity, and each guinea pig has a counter starting from $0$. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each $1\leq i\leq n$ we have that the $i-$th guinea pig has $i-1$ in its counter.

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$. Prove that points $P,Q,P_1$, and $Q_1$ are concyclic. [i]Proposed by Anton Trygub, Ukraine[/i]

Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$. a) Show that $super sextalternados$ numbers don't exist. b) Find the smallest $sextalternado$ number.

Find the maximum possible number of kings on a $12\times 12$ chess table so that each king attacks exactly one of the other kings (a king attacks only the squares that have a common point with the square he sits on).

Let $t = 2016$ and $p = \ln 2$. Evaluate in closed form the sum \[ \sum_{k=1}^{\infty} \left( 1-\sum_{n=0}^{k-1}\frac{e^{-t}t^{n}}{n!} \right) \left(1-p\right)^{k-1}p. \]

Let $n > 2$ be an integer. Show that it is possible to choose $n$ points in the plane, not all of them lying on the same line, such that the distance between any pair of points is an integer (that is, $\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}$ is an integer for all pairs $(x_1, y_1)$ and $(x_2, y_2)$ of points).

Let $\{x_n\}_n$ be a sequence of positive rational numbers, such that $x_1$ is a positive integer, and for all positive integers $n$. $x_n = \frac{2(n - 1)}{n} x_{n-1}$, if $x_{n_1} \le 1$ $x_n = \frac{(n - 1)x_{n-1} - 1}{n}$ , if $x_{n_1} > 1$. Prove that there exists a constant subsequence of $\{x_n\}_n$.

A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).

Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ , and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$ , determine the largest possible value of $y$.

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

Consider a positive integer $n$ and two points $A$ and $B$ in a plane. Starting from point $A$, $n$ rays and starting from point $B$, $n$ rays are drawn so that all of them are on the same half-plane defined by the line $AB$ and that the angles formed by the $2n$ rays with the segment $AB$ are all acute. Define circles passing through points $A$, $B$ and each meeting point between the rays. What is the smallest number of [b]distinct [/b] circles that can be defined by this construction?

Let $k\ge 3$ be an integer. Prove that if $n>\binom k3$, then for any $3n$ pairwise different real numbers $a_i,b_i,c_i$ ($1\le i\le n$), among the numbers $a_i+b_i$, $a_i+c_i$, $b_i+c_i$, one can find at least $k+1$ pairwise different numbers. Show that this is not always the case when $n=\binom k3$.

What are the last two digits of $$2003^{{2005}^{{2007}^{2009}}}$$ , where $a^{b{^c}}$ means $a^{(b^c)}$?

Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$.

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$. Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$.

Prove that for any natural numbers $n,r$ with $r + 3 \le n $the binomial coefficients $n \choose r$, $n \choose r+1$, $n \choose r+2 $, $n \choose r+3 $ cannot be successive terms of an arithmetic progression.

Let $a$ and $b$ be the positive integers. Show that at least one of the numbers $a$, $b$, $a+b$ can be expressed as the difference of the squares of two integers.

Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.

For a nonzero integer $i$, the exponent of $2$ in the prime factorization of $i$ is called $ord_2 (i)$. For example, $ord_2(9)=0$ since $9$ is odd, and $ord_2(28)=2$ since $28=2^2\times7$. The numbers $3^n-1$ for $n=1,2,3,\ldots$ are all even so $ord_2(3^n-1)>0$ for $n>0$. a) For which positive integers $n$ is $ord_2(3^n-1) = 1$? b) For which positive integers $n$ is $ord_2(3^n-1) = 2$? c) For which positive integers $n$ is $ord_2(3^n-1) = 3$? Prove your answers.

Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]

Let $A,B,C,D$ and $E$ be five points lying in this order on a circle, such that $AD=BC$. The lines $AD$ and $BC$ meet at a point $F$. The circumcircles of the triangles $CEF$ and $ABF$ meet again at the point $P$. Prove that the circumcircles of triangles $BDF$ and $BEP$ are tangent to each other.

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.