Finite number of dwarfs excavates ore in the mine with infinite number of levels. Each day at the same time one dwarf from each level, inhabited with exactly $n = 2, 3, ... $ dwarfs, move $n-1$ levels down. Prove that after some moment there will be no more then one dwarf on each level.

Let $ABC$ be a triangle, and let $B_1$ and $B_2$ be points on segment $AC$ symmetric with respect to the midpoint of $AC$. Let $\gamma_A$ denote the circle passing through $B_1$ and tangent to line $AB$ at $A$. Similarly, let $\gamma_C$ denote the circle passing through $B_1$ and tangent to line $BC$ at $C$. Let the circles $\gamma_A$ and $\gamma_C$ intersect again at point $B'$ ($B' \neq B_1$). Prove that $\angle ABB' = \angle CBB_2$.

Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$. [b]i.[/b] Prove that the lines $PR,ST$ intersect at $A$. [b]ii.[/b] Prove that the lines $PR\cap MD$ intersect on $c$. [asy]import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); draw((0.,0.)--(5.,0.), uququq); draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); draw((0.6699432366054657,0.)--(5.,3.257603675597893)); draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); /* dots and labels */ dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); label("$A$", (0.7472169504504719,2.65), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.2,-0.4), NE * labelscalefactor); dot((5.,0.),linewidth(3.pt) + dotstyle); label("$C$", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); dot((2.5,0.),linewidth(3.pt) + dotstyle); label("$D$", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$E$", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$Z$", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); label("$K$", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); label("$P$", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); label("$S$", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); label("$T$", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); label("$R$", (3.445433497126431,4.375025554412117), NE * labelscalefactor); dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); label("$M$", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims: (i) Less than $12$ employees have a more difficult work; (ii) At least $30$ employees take a higher salary. Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.

Given an equilateral triangle lattice composed of $\frac{n(n+1)}{2}$ points (as shown in the figure), record the number of equilateral triangles with three points in the lattice as vertices as $f(n)$. Find an expression for $f(n)$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/1de27231e8ef9c1c6a3dfd590a7c71adc508d6.png[/img]

Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.

In a quadrilateral $ABCD$ diagonal $AC$ is a bisector of $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X$ and $Y$ are the feet of the perpendiculars from $A$ to $BC$ and $CD$ respectively. Prove that the orthocenter of $\triangle AXY$ lies on the line $BD$.

Let $ABCD$ be a square, and let $l$ be a line passing through the midpoint of segment $AB$ that intersects segment $BC$. Given that the distances from $A$ and $C$ to $l$ are $4$ and $7$, respectively, compute the area of $ABCD$.

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display? $ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$

A quadrangle was drawn on the board, that you can inscribe and circumscribe a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.

The boba shop sells four different types of milk tea, and William likes to get tea each weekday. If William refuses to have the same type of tea on successive days, how many different combinations could he get, Monday through Friday?

How many polynomial functions $f$ of degree $\ge 1$ satisfy \[ f(x^2)=[f(x)]^2=f(f(x)) \ ? \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{finitely many but more than 2} \\ \qquad\textbf{(E)}\ \text{infinitely many} $

Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that $$xf(x)+(f(y))^2+2xf(y)$$ is perfect square for all positive integers $x,y$. **This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.

The number of real values of $x$ that satisfy the equation\[ (2^{6x+3})(4^{3x+6})=8^{4x+5} \]is: $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{greater than 3}$

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

Given two circles $k_1$ and $k_2$ which intersect at $Q$ and $Q'.$ Let $P$ be a point on $k_2$ and inside of $k_1 $ such that the line $PQ$ intersects $k_1$ in a point $X\ne Q$ and such that the tangent to $k_1$ at $X$ intersects $k_2$ in points $A$ and $B.$ Let $k$ be the circle through $A,B$ which is tangent to the line through $P$ parallel to $AB.$ Prove that the circles $k_1$ and $k$ are tangent.

Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.

The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.

Hexagon $ABCDEF$ has a circumscribed circle and an inscribed circle. If $AB = 9$, $BC = 6$, $CD = 2$, and $EF = 4$. Find $\{DE, FA\}$.

Two circles each have radius 1. No point is inside both circles. The circles are contained in a square. What is the area of the smallest such square?

There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

Jason and Valerie agree to meet for game night, which runs from $4:00$ PM to $5:00$ PM. Jason and Valerie each choose a random time from $4:00$ PM to $5:00$ PM to show up. If Jason arrives first, he will wait $20$ minutes for Valerie before leaving. If Valerie arrives first, she will wait $10$ minutes for Jason before leaving. What is the probability that Jason and Valerie successfully meet each other for game night?