Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.

Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.

Let $ n \in N $. Let's define $ S_n = \{1, ..., n \} $. Let $ x_1 <x_2 <\cdots <x_n $ be any real. Determine the largest possible number of pairs $ (i, j) \in S_n \times S_n $ with $ i \not = j $, for which it is true that $ 1 <| x_i-x_j | <2 $ and justify why said value cannot be higher.

Let $a_0=0$, $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers. (i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which $$p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k$$ and $$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)$$ where the constant $c$ depends only on the numbers $a_i, b_i$. (ii) Prove that, in general, (*) cannot be replaced by the relation $$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$ [J. Szabados]

Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral . (G . Galperin , A . Savin, Moscow)

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? $ \text{(A)}\ 2\pi/3 \qquad \text{(B)}\ 2\pi \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi$

Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country

For all numbers $n\ge 1$ does there exist infinite positive numbers sequence $x_1,x_2,...,x_n$ such that $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_n}$

Alexis has $2020$ paintings, the $i$th one of which is a $1\times i$ rectangle for $i = 1, 2, \ldots, 2020$. Compute the smallest integer $n$ for which they can place all of the paintings onto an $n\times n$ mahogany table without overlapping or hanging off the table. [i]Proposed by Brandon Wang[/i]

We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?

Let $ABCD$ be a trapezoid with base sides $AB$ and $CD$ and let $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=m$ and $BD=n$. We know that $m^2+n^2=(a+c)^2$ $a)$ Prove that lines $AC$ and $BD$ are perpendicular $b)$ Prove that $ac<bd$

Consider the parabola $ P:x-y^2-(p+3)y-p=0,p\in\mathbb{R}^*. $ Show that $ P $ intersects the coordonate axis at three points, and that the circle formed by these three points passes through a fixed point.

Let $a,b,c$ be real numbers greater than $1$. Prove the inequality $$\log_a\left(\frac{b^2}{ac}-b+ac\right)\log_b\left(\frac{c^2}{ab}-c+ab\right)\log_c\left(\frac{a^2}{bc}-a+bc\right)\ge1.$$

Two sets $S_1$ and $S_2$, which are not necessarily distinct, are each selected randomly and independently from each other among the $512$ subsets of $S = \{1, 2, \ldots ,9\}$. Let $\sigma(X)$ denote the sum of the elements of set $X$. Note that $\sigma(\emptyset) = 0$ where $\emptyset$ denotes the empty set. If $S_1 \cup S_2$ stands for the union of $S_1$ and $S_2$, the probability that $\sigma(S_1 \cup S_2)$ is divisible by $3$ can be expressed as a common fraction of the form $\tfrac{m}{2^n}$ where $m$ is odd and $n$ is a positive integer. Find $m + n$.

Let $p$ be an odd prime integer and $k$ a positive integer not divisible by $p$, $1\le k<2(p+1)$, and let $N=2kp+1$. Prove that the following statements are equivalent: (i) $N$ is a prime number (ii) there exists a positive integer $a$, $2\le a<n$, such that $a^{kp}+1$ is divisible by $N$ and $\gcd\left(a^k+1,N\right)=1$.

Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. Then, which two of the following statements are true? $ \text{(1)}\ \text{the product MD cannot be less than abc} \qquad$ $\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad$ $\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad$ $\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}$ $\text{ (This means: no two have a common factor greater than 1.)}$ $ \textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $

For any permutation $p$ of set $\{1, 2, \ldots, n\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$

The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

Find all $4$-digit numbers $\overline{abba}$ that are equal to the product of some consecutive prime numbers.

Side $BC$ of triangle $ABC$ is equal to $a$ and the opposite angle is equal to $\theta$. A straight line passing through the midpoint of $BC$ and the center of the inscribed circle intersects lines $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the quadrilateral (non-convex) $BMPC$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

In order to determine $\frac{1}{A}$ for $A>0$, one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$

Three of the $16$ squares from a $4 \times 4$ grid of squares are selected at random. The probability that at least one corner square of the grid is selected is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $