In a room, we have 2019 aligned switches, connected to 2019 light bulbs, all initially switched on. Then, 2019 people enter the room one by one, performing the operation: The first, uses all the switches; the second, every second switch; the third, every third switch, and so on. How many lightbulbs remain switched on, after all the people entered ?

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$ [asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy] $\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)$

If $X_i$ is the answer to problem $i$ for $1 \le i \le 12$, find the minimum possible value of $\sum_{n=1}^{12} (-1)^n X_n$. [i]Proposed by Evan Chen, Lewis Chen[/i]

Given a real number $\alpha$ satisfying $\alpha^3 = \alpha + 1$. Determine all $4$-tuples of rational numbers $(a, b, c, d)$ satisfying: $a\alpha^2 + b\alpha+ c = \sqrt{d}.$

A bag contains plastic cubes of the same size, whose faces have been painted in colors: white, red, yellow, green, blue and violet (without repeating a color on two faces of the same cube). How many of these cubes can there be distinguishable to each other?

We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin? [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

[b]Q4.[/b] A man travels from town $A$ to town $E$ through $B,C$ and $D$ with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town $A$ and town $E$ can be classified as horizontal, up slope, down slope and horizontal and total length of each typr of road is the same, what is the average speed of his journey? \[(A) \; 2 \text{km/h} \qquad (B) \; 2,5 \text{km/h} ; \qquad (C ) \; 3 \text{km/h} ; \qquad (D) \; 3,5 \text{km/h} ; \qquad (E) \; 4 \text{km/h}.\]

A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root. Find all possible values of $P(0)$

Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days? ${ \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$

Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$

Let $A_1A_2A_3\cdots A_{10}$ be a regular decagon and $A=A_1A_4\cap A_2A_5, B=A_1A_6\cap A_2A_7, C=A_1A_9\cap A_2A_{10}.$ Find the angles of the triangle $ABC$.

Given a set of $1997$ numbers such that if each number in the set, replace with the sum of the rest, you get the same set. Prove that the product of numbers in the set is equal to $0$.

What is the maximum number of bishops that can be placed on an $ 8 \times 8 $ chessboard such that at most three bishops lie on any diagonal?

Let $A_1A_2\ldots A_6$ be a regular hexagon with side length $11\sqrt{3},$ and let $B_1B_2\ldots B_6$ be another regular hexagon completely inside $A_1A_2\ldots A_6$ such that for all $i \in \{1, 2, \ldots, 5\}$ $A_iA_{i+1}$ is parallel to $B_iB_{i+1}.$ Suppose that the distance between lines $A_1A_2$ and $B_1B_2$ is $7,$ the distance between lines $A_2A_3$ and $B_2B_3$ is $3,$ and the distance between lines $A_3A_4$ and $B_3B_4$ is $8.$ Compute the side length of $B_1B_2\ldots B_6.$

Suppose there are 8 white balls and 2 red balls in a packet. Each time one ball is drawn and replaced by a white one. Find the probability that the last red ball is drawn in the fourth draw.

If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.

Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.

Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.

Prove that for a natural number $n$ greater than 1, the following conditions are equivalent: a) $ n $ is an even number, b) there is a permutation $ (a_0, a_1, a_2, \ldots, a_{n-1}) $ of the set $ \{0,1,2,\ldots,n—1\} $ with the property that the sequence of residues from dividing by $ n $ the numbers $ a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots, a_0 + a_1 + a_2 + \ldots a_{n-1} $ is also a permutation of this set.

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.

(1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$. (2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)f(x)dx\equal{}3\sqrt{3}$, then we have $ f(x)\equal{}\boxed{\ A\ }(x\minus{}1)\plus{}\boxed{\ B\ }(x\plus{}1)$, thus we have $ f(x)\equal{}\boxed{\ C\ }$.

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$. a) Prove that the points $P, Q$, and $R$ are collinear. b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.

Let $p$ be a prime number gretater then $3$. What is the number of pairs $(m, n)$ of integers with $0 <m <n <p$, for which the polynomial $x^p + px^n + px^m +1$ is not a product of two non-constant polynomials with integer coefficients can be written?

Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?