Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

Let $ ABC $ be an inscribed triangle in circle $ (O) $. Let $ D $ be the intersection of the two tangent lines of $ (O) $ at $ B $ and $ C $. The circle passing through $ A $ and tangent to $ BC $ at $ B $ intersects the median passing $ A $ of the triangle $ ABC $ at $ G $. Lines $ BG, CG $ intersect $ CD, BD $ at $ E, F $ respectively. a) The line passing through the midpoint of $ BE $ and $ CF $ cuts $ BF, CE $ at $ M, N $ respectively. Prove that the points $ A, D, M, N $ belong to the same circle. b) Let $ AD, AG $ intersect the circumcircle of the triangles $ DBC, GBC $ at $ H, K $ respectively. The perpendicular bisectors of $ HK, HE$, and $HF $ cut $ BC, CA$, and $AB $ at $ R, P$, and $Q $ respectively. Prove that the points $ R, P$, and $Q $ are collinear.

Is there a described $n$-gon in which each side is longer than the diameter of the inscribed circle a) at $n = 4$? b) when $n = 7$? c) when $n = 6$? [i]P.A.Kozhevnikov[/i]

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ . Proposed by Alireza Dadgarnia

$AB < AC$ on $\triangle ABC$. The midpoint of arc $BC$ which doesn't include $A$ is $T$ and which includes $A$ is $S$. On segment $AB,AC$, $D,E$ exist so that $DE$ and $BC$ are parallel. The outer angle bisector of $\angle ABE$ and $\angle ACD$ meets $AS$ at $P$ and $Q$. Prove that the circumcircle of $\triangle PBE$ and $\triangle QCD$ meets on $AT$.

Let $O$ be the circumcenter of $\triangle ABC$. A circle with center $P$ pass through $A$ and $O$ and $OP$//$BC$. $D$ is a point such that $\angle DBA = \angle DCA = \angle BAC$. Prove that: Circle $(P)$, circle $(BCD)$ and the circle with diameter $(AD)$ share a common point. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS9jLzlmZjdlN2ExZDJjYjAwYWJlZTQzYWRkYzg3NDlhMTUyZjRlNGJjLmpwZw==&rn=c291dGhlYXN0UDYuanBn[/img]

Let $ABC$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $BC$ such that $BE = EF = F C$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, so that $X$, $Y$ , and $G$ are not collinear. If the line through $E$ parallel to $XG$ and the line through $F$ parallel to $Y G$ intersect at $P\ne G$, prove that $GP$ passes through the midpoint of $XY$.

Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$. [i](Karl Czakler)[/i]

[u]Round 1[/u] [b]p1.[/b] Ash is running around town catching Pokémon. Each day, he may add $3, 4$, or $5$ Pokémon to his collection, but he can never add the same number of Pokémon on two consecutive days. What is the smallest number of days it could take for him to collect exactly $100$ Pokémon? [b]p2.[/b] Jack and Jill have ten buckets. One bucket can hold up to $1$ gallon of water, another can hold up to $2$ gallons, and so on, with the largest able to hold up to $10$ gallons. The ten buckets are arranged in a line as shown below. Jack and Jill can pour some amount of water into each bucket, but no bucket can have less water than the one to its left. Is it possible that together, the ten buckets can hold 36 gallons of water? [img]https://cdn.artofproblemsolving.com/attachments/f/8/0b6524bebe8fe859fe7b1bc887ac786106fc17.png[/img] [b]p3.[/b] There are $2023$ knights and liars standing in a row. Knights always tell the truth and liars always lie. Each of them says, “the number of liars to the left of me is greater than the number of knights to the right.” How many liars are there? [b]p4.[/b] Camila has a deck of $101$ cards numbered $1, 2, ..., 101$. She starts with $50$ random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all $50$ cards in her hand with her choice of $50$ of the $51$ cards from the table. Show that Camila can make at most 50 exchanges and end up with cards $1, 2, ..., 50$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.png[/img] [b]p5.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [u]Round 2[/u] [b]p6.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company will lay lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses. The Edison lighting company will hang strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [b]p7.[/b] You are given a sequence of $16$ digits. Is it always possible to select one or more digits in a row, so that multiplying them results in a square number? [img]https://cdn.artofproblemsolving.com/attachments/d/1/f4fcda2e1e6d4a1f3a56cd1a04029dffcd3529.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the needle leg could be positioned at any angle with respect to the paper. Let $n$ be the difference between the radii of the largest and the smallest circles that Nadia can draw with this compass in centimeters. Which of the following options is closest to $n$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 20$

An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

The diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$ intersect at a point O. It is known that $\angle BAD = 60^o$ and $AO = 3OC$. Prove that the sum of some two sides of a quadrilateral is equal to the sum of the other two sides.

Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.

Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.

$N \ge 3$ different points are marked on the plane. It is known that among pairwise distances between marked points there are not more than $n$ different distances. Prove that $N \le (n + 1)^2$.

Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$? $ \textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}\qquad \textbf{(D)}\ 35\sqrt{3}\qquad \textbf{(E)}\ 36\sqrt{3}$

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

In an acute triangle $ABC$, the feet of the perpendiculars from $A$ and $C$ to the opposite sides are $D$ and $E$, respectively. The line passing through $E$ and parallel to $BC$ intersects $AC$ at $F$, the line passing through $D$ and parallel to $AB$ intersects $AC$ at $G$. The feet of the perpendiculars from $F$ to $DG$ and $GE$ are $K$ and $L$, respectively. $KL$ intersects $ED$ at $M$. Prove that $FM \perp ED$.

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$. Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively Given that $EM = FM$, compute $\angle EMF$.

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$. [i]Proposed by Anant Mudgal[/i]

A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.