Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

Four congruent semicircular half-disks are arranged inside a circle with radius $4$ so that each semicircle is internally tangent to the circle, and the diameters of the semicircles form a $2\times 2$ square centered at the center of the circle as shown. The radius of each semicircular half-disk is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/e/8c0b9fdd69f6b54d39708da94ef2b2d039cb1e.png[/img]

MTRPia in $2044$ is highly advanced and a lot of the work is done by disc-shaped robots, each of radius $1$ unit. In order to not collide with each other, there robots have a smaller $360$-degree camera mounted on top, as shown in the figure (robot $r_1$ 'sees' robot $r_2$). Each of there cameras themselves are smaller discs of radius $c$. Suppose there are three robots $r_1, r_2, r_3$ placed 'consecutively' such that $r_2$ is roughly in the middle. The angle between the lines joining the centres of $r_1, r_2$ and $r_2, r_3$ is given to be $\theta$. The distance between the centres of $r_1,r_2 = $ distance between centres of $r_2,r_3 = d$. Show (with the aid of clear diagrams) that $r_1$ and $r_3$ can see each other iff $\sin{\theta} > \frac{1-c}{d}$. As a bonus, try to show that in a longer 'chain' of such robots (same $d$, $\theta$), if $\sin{\theta} > \frac{1-c}{d}$ then all robots can see each other.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

In triangle $ABC$, medians $AA'$, $BB'$, $CC'$ are extended until they intersect with the circumcircle at points $A_0$, $B_0$, $C_0$, respectively. It is known that the intersection point M of the medians of triangle $ABC$ divides the segment $AA_0$ in half. Prove that the triangle $A_0B_0C_0$ is isosceles.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D.1 / L.1[/b] Find the units digit of $3^{1^{3^{3^7}}}$. [b]D.2[/b] Find the positive solution to the equation $x^3 - x^2 = x - 1$. [b]D.3[/b] Points $A$ and $B$ lie on a unit circle centered at O and are distance $1$ apart. What is the degree measure of $\angle AOB$? [b]D.4[/b] A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between $1$ and $2019$, inclusive? [b]D.5[/b] Ted has four children of ages $10$, $12$, $15$, and $17$. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now? [u]Set 2[/u] [b]D.6[/b] Mr. Schwartz is on the show Wipeout, and is standing on the first of $5$ balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is $1/2$, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish. [b]D.7 / L. 5[/b] Kevin has written $5$ MBMT questions. The shortest question is $5$ words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question? [b]D.8 / L. 3[/b] Square $ABCD$ with side length $1$ is rolled into a cylinder by attaching side $AD$ to side $BC$. What is the volume of that cylinder? [b]D.9 / L.4[/b] Haydn is selling pies to Grace. He has $4$ pumpkin pies, $3$ apple pies, and $1$ blueberry pie. If Grace wants $3$ pies, how many different pie orders can she have? [b]D.10[/b] Daniel has enough dough to make $8$ $12$-inch pizzas and $12$ $8$-inch pizzas. However, he only wants to make $10$-inch pizzas. At most how many $10$-inch pizzas can he make? [u]Set 3[/u] [b]D.11 / L.2[/b] A standard deck of cards contains $13$ cards of each suit (clubs, diamonds, hearts, and spades). After drawing $51$ cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs? [b]D.12 / L. 7[/b] Let $s(n)$ be the sum of the digits of $n$. Let $g(n)$ be the number of times s must be applied to n until it has only $1$ digit. Find the smallest n greater than $2019$ such that $g(n) \ne g(n + 1)$. [b]D.13 / L. 8[/b] In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the $2019$ contestants, is secretly told that her score is $S$. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of $S$? [b]D.14 / L. 9[/b] Let $A$ and $B$ be opposite vertices on a cube with side length $1$, and let $X$ be a point on that cube. Given that the distance along the surface of the cube from $A$ to $X$ is $1$, find the maximum possible distance along the surface of the cube from $B$ to $X$. [b]D.15[/b] A function $f$ with $f(2) > 0$ satisfies the identity $f(ab) = f(a) + f(b)$ for all $a, b > 0$. Compute $\frac{f(2^{2019})}{f(23)}$. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.

Consider an isosceles triangle $ABC$ with $AB=AC$. Point $D$ is on the smaller arc $AB$ of its circumcirle. Point $E$ lies on the continuation of segment $AD$ beyond point $D$ so that both $A$ and $E$ lie in the same half-plane relative to $BC$. The circumcirle of triangle $BDE$ intersects side $AB$ at point $F$. Prove that lines $EF$ and $BC$ are parallel. (Author: R. Zhenodarov)

Let $\triangle ABC$ be an acute-angled triangle and $A_1, B_1$, and $C_1$ be the feet of the altitudes from $A, B$, and $C$, respectively. On the rays $AA_1, BB_1$, and $CC_1$, we have points $A_2, B_2$, and $C_2$ respectively, lying outside of $\triangle ABC$, such that \[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\] If the intersections of $B_1C_2$ and $B_2C_1$, $C_1A_2$ and $C_2A_1$, and $A_1B_2$ and $A_2B_1$ are $A', B'$, and $C'$ respectively, prove that $AA', BB'$, and $CC'$ have a common point.

The circle $\omega$ lies inside the circle $\Omega$ and touches it internally at $T.$ Let $XY{}$ be a variable chord of the circle $\Omega$ touching $\omega.$ Denote by $X'$ and $Y'$ the midpoints of the arcs $TY{}$ and $TX{}$ which do not contain $X{}$ and $Y{}$ respectively. Prove that all possible lines $X'Y'$ pass through a fixed point. [i]Proposed by F. Petrov[/i]

Circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects the circle $k_1$ at points $A$ and $C$, the circle $K_2$ at points $B$ and $D$ so that the points $A,B,C$ and $D$ lie on the line $\ell$ are in that order. Let $X$ a point on the line $MN$ such that the point $M$ is located between the points $X$ and $N$. Let $P$ be the intersection of lines $AX$ and $BM$, and $Q$ be the intersection of lines $DX$ and $CM$. If $K$ is the midpoint of segment $AD$ and $L$ is the midpoint of segment $BC$, prove that the lines $XK$ and $ML$ intersect on the line $PQ$.

One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is $ \textbf{(A)}\ 600 \qquad \textbf{(B)}\ 520 \qquad \textbf{(C)}\ 488 \qquad \textbf{(D)}\ 480 \qquad \textbf{(E)}\ 400$

Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that: (i) $f (r)$ is a line for every line $r$, (ii) $f (r) $ is parallel to $r$ for every line $r$. What possible transformations can $f$ be?

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation \[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \] where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that \[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\] (b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.

Two different points $A$ and $B$ have been marked on the circle $\omega$. We consider all points $X$ of the circle $\omega$, different from $A$ and $B$. Let $Y$ be the middpoint of the chord $AX$ and $Z$ be the projection of point $A$ on the line $BX$. Prove that all straight lines $YZ$ pass through a certain fixed point that does not depend on the choice of point $X$.

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

Let $ABCD$ be a cyclic quadrilateral such that the lines $AB$ and $CD$ intersects in $K$, let $M$ and $N$ be the midpoints of $AC$ and $CK$ respectively. Find the possible value(s) of $\angle ADC$ if the quadrilateral $MBND$ is cyclic.

$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.

$ABC$ is an acute-angled triangle. Let $D$ be the point on $BC$ such that $AD$ is the bisector of $\angle A$. Let $E, F$ be the feet of perpendiculars from $D$ to $AC,AB$ respectively. Suppose the lines $BE$ and $CF$ meet at $H$. The circumcircle of triangle $AFH$ meets $BE$ at $G$ (apart from $H$). Prove that the triangle constructed from $BG$, $GE$ and $BF$ is right-angled.

There are $100$ points of general position marked on the plane (i.e. no three lie on the same straight line). Prove that it is possible to select three marked points $A, B, C$ so that for any point $D$ of the remaining $97$ marked points, the lines $AD$ and $CD$ would not contain points lying inside the triangle $ABC$.

Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$. [i]Italy[/i]

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

Given are $5$ segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.

$2021$ points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least $673$ of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)

Let $ABC$ be a triangle, let $A_1, B_1, C_1$ be the antipodes of the vertices $A, B, C$, respectively, in the circle $ABC$, and let $X$ be a point in the plane $ABC$, collinear with no two vertices of the triangle $ABC$. The line through $B$, perpendicular to the line $XB$, and the line through $C$, perpendicular to the line $XC$, meet at $A_2$, the points $B_2$ and $C_2$ are defined similarly. Show that the lines $A_1A_2, B_1B_2$ and $C_1C_2$ are concurrent.