Circles $Q_1$ and $Q_2$ are tangent to each other externally at $T$. Points $A$ and $E$ are on $Q_1$, lines $AB$ and $DE$ are tangent to $Q_2$ at $B$ and $D$, respectively, lines $AE$ and $BD$ meet at point $P$. Prove that (1) $\frac{AB}{AT}=\frac{ED}{ET}$; (2) $\angle ATP + \angle ETP = 180^{\circ}$. [asy]import graph; size(5.97cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-6,xmax=5.94,ymin=-3.19,ymax=3.43; pair Q_1=(-2.5,-0.5), T=(-1.5,-0.5), Q_2=(0.5,-0.5), A=(-2.09,0.41), B=(-0.42,1.28), D=(-0.2,-2.37), P=(-0.52,2.96); D(CR(Q_1,1)); D(CR(Q_2,2)); D(A--B); D((-3.13,-1.27)--D); D(P--(-3.13,-1.27)); D(P--D); D(T--(-3.13,-1.27)); D(T--A); D(T--P); D(Q_1); MP("Q_1",(-2.46,-0.44),NE*lsf); D(T); MP("T",(-1.46,-0.44),NE*lsf); D(Q_2); MP("Q_2",(0.54,-0.44),NE*lsf); D(A); MP("A",(-2.22,0.58),NE*lsf); D(B); MP("B",(-0.35,1.45),NE*lsf); D((-3.13,-1.27)); MP("E",(-3.52,-1.62),NE*lsf); D(D); MP("D",(-0.17,-2.31),NE*lsf); D(P); MP("P",(-0.47,3.02),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

Let $\mathcal{P}$ be a convex polygon and $\textbf{T}$ be a triangle with vertices among the vertices of $\mathcal{P}$. By removing $\textbf{T}$ from $\mathcal{P}$, we end up with $0, 1, 2,$ or $3$ smaller polygons (possibly with shared vertices) which we call the effect of $\textbf{T}$. A triangulation of $P$ is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of $\mathcal{P}$ $\underline{\text{beautiful}}$, if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of $\mathcal{P}$ is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]

For how many pairs of integers $(m, n)$ with $0 < m \le n \le 50$ do there exist precisely four triples of integers $(x, y, z)$ satisfying the following system? \[\begin{cases} x^2 + y+ z = m \\ x + y^2 + z = n\end{cases}\] \[\rm a. ~180\qquad \mathrm b. ~182\qquad \mathrm c. ~186 \qquad\mathrm d. ~188\qquad\mathrm e. ~190\]

Determine the quantity of positive integers $N$ of $10$ digits with the following properties: I- All the digits of $N$ are non-zero. II- $11|N$. III- $N$ and all the permutation(s) of the digits of $N$ are divisible by $12$.

Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

Let $\omega_1$ and $\omega_2$ be intersecting circles in the plane with radii $12$ and $15$, respectively. Suppose $\Gamma$ is a circle such that $\omega_1$ and $\omega_2$ are internally tangent to $\Gamma$ at $X_1$ and $X_2$, respectively. Similarly, $\ell$ is a line that is tangent to $\omega_1$ and $\omega_2$ at $Y_1$ and $Y_2$, respectively. If $X_1X_2=18$ and $Y_1Y_2=9$, what is the radius of $\Gamma$?

What is the period of the function $f(x)=\cos(\cos(x))$?

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.

What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?

$N$ straight lines are drawn on a plane. The $N$ lines can be partitioned into set of lines such that if a line $l$ belongs to a partition set then all lines parallel to $l$ make up the rest of that set. For each $n>=1$,let $a_n$ denote the number of partition sets of size $n$. Now that $N$ lines intersect at certain points on the plane. For each $n>=2$ let $b_n$ denote the number of points that are intersection of exactly $n$ lines. Show that $\sum_{n>= 2}(a_n+b_n)$$\binom{n}{2}$ $=$ $\binom{N}{2}$

Let $r$ be a positive integer. Find the smallest positive integer $m$ satisfying the condition: For all sets $A_1, A_2, \dots, A_r$ with $A_i \cap A_j = \emptyset$, for all $i \neq j$, and $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$, there exists $a, b \in A_k$ for some $k$ such that $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$.

For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.