Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$. Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.

The area of a circle whose circumference is $ 24\pi$ is $ k\pi$. What is the value of $ k$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 144$

An acute triangle is a triangle that has all angles less that $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be a right-angled triangle with $\angle ACB =90^{\circ}.$ Let $CD$ be the altitude from $C$ to $AB,$ and let $E$ be the intersection of the angle bisector of $\angle ACD$ with $AD.$ Let $EF$ be the altitude from $E$ to $BC.$ Prove that the circumcircle of $BEF$ passes through the midpoint of $CE.$

Let $ABC$ a triangle and $k$ a circle such that: (1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$ (2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$ (3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$ (4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$ Prove that the line $XY$ is tangent to the circumcircle of $BQY.$

In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC, D\neq B$ and $D\neq C$. The circle $ABD$ meets the segment $AC$ again at an interior point $E$. The circle $ACD$ meets the segment $AB$ again at an interior point $F$. Let $A'$ be the reflection of $A$ in the line $BC$. The lines $A'C$ and $DE$ meet at $P$, and the lines $A'B$ and $DF$ meet at $Q$. Prove that the lines $AD, BP$ and $CQ$ are concurrent (or all parallel).

Let $ABC$ be a right-angled triangle in $A$ such that $AB<AC$. Let $M$ be the midpoint of $BC$ and let $D$ be the intersection of $AC$ with the perpendicular line to $BC$ which passes through $M$. Let $E$ be the intersection point of the parallel line to $AC$ which passes through $M$ with the perpendicular line to $BD$ which passes through $B$. Prove that triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC=60^{\circ}$.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$. Prove that $\angle BXP = \angle CXQ$. [i]Kian Moshiri, United Kingdom[/i]

The trapezoid is composed of three conguent right isosceles triangles as shown in the figure. It is necessary to cut it into $4$ equal parts. How to do it? [img]https://cdn.artofproblemsolving.com/attachments/f/e/87b07ae823190f26b70bfa22824679a829e649.png[/img]

A [i]hedgehog[/i] is a circle without its boundaries. The diameter of the hedgehog is the diameter of the corresponding circle. We say that the hedgehog sits at the at the point where the center of the circle is located. We are given a triangle with sides $a, b, c$, with hedgehogs sitting at its vertices. It is known that inside the triangle there is a point from which you can reach any side of the triangle by walking along a straight line without hitting any hedgehog. What is the largest possible sum of the diameters of these hedgehogs? [i]Proposed by Oleksiy Masalitin[/i]

Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

[u]Round 1[/u] [b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!) [b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started? [b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips? [u]Round 2[/u] [b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine? [b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved? [b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$. [u]Round 3 [/u] [b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$. [b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that $\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$. $\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over. Find the smallest possible number of boxes of food. [b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$. [u]Round 4 [/u] [b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$. [b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$? [b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $

Let $HOPKINS$ be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than $180^{\circ}$) with area $1876$ such that all of its side lengths are greater than $5$, $OP=20$, and $KI=22$. Arcs with radius $2$ are drawn inside $HOPKINS$ with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside $HOPKINS$ but outside the sectors.

Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.

In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$. (a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle. (b) Prove that the line $ST$ is parallel to the side $BC$. (Karl Czakler)

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

A fudgeflake is a planar fractal figure with a $120^{\circ}$ rotational symmetry such that three identical fudgeflakes in the same orientation fit together without gaps to form a larger fudgeflake with its orientation $30^{\circ}$ clockwise of the smaller fudgeflakes' orientation, as shown below. If the distance between the centers of the original three fudgeflakes is $1$, what is the area of one of those three fudgeflakes? Justify your answer. [asy] defaultpen(linewidth(.7)); string s = "00d08c8520612022202288272220065886,00e0708768822888788866683,01006c8765,01206b88227606,01208c858768588616678868160027,017068870228728868822872272220600278886,0190988886872272882228166060,02209486868506,02304b282022852282022828888828228166060066002000668666,02304d8882858688886166702,023050,0230637222282272220786,0240918681,02505f,0260908527222027285886852728522820766,02905b81,03904422888888766686882288888607,03908c58588685876668688228882078688228822220258587685886166702,049053852282000027888666,0490fa852061112222282282000702202220065868,04a0ae868822888888866602768688228888860728228822820228586888766702,04a0de2821666868822888500602,04b0ac5812022882227200070,04c04a8220228822272012882876882288227606,04c0da288282220228886882288227600660020060668,04c0f2,04e0fa88868588616668688228882078688228822220786,05e0ba878605,05e0d287688872006,05e102786720,05f0b88220228822816606066860,06002786882288886888666868886166600668666868887,0600748207,0600ce88616,0600ff86816,0610258222282220228887882288228202285868886668688228858688228822827066,062023223,0620cd88227600,0620fe8527222027686,064074555555555555555555555555555555555552882288888876668688866606060276866686882288878886668668522888868527282822816660222228588688861668678886166600668666868886660602782228,0650c985877,0680b0865282888882822202288878886668678282858768522882822200602788860660606,06a0fa88886070220,06c0b48830,0700fe8527222206702,0790fb867888666868822888788228722722720128288768886668688866600668666027686882288886888606786882288888607,0870b9822202288222022888886058,0880428582288868886668688228828768822882282166606060602,088070816,08903e7870,08a06c877,08b02a82858868886067222006,08b03b867202285272220786,08c06b8528220676,09003572,0900745555555555555555555555555555555555555555555555555555555555555555555555555558207,0920668858888285272220786,09606c,09812e88228166,09a128868827,09b12682202288222766060061,09c0648867,09c094868672020228160,09d04b86787228828868886668688866616006686605820228,09d0e38207,09e0908282822202288888828282220228886888681666868522888888285228202282886685886066886888606606666678868160027,09f0478765,0a10468822220276866606,0a411e828886882288227606060602,0a90428888681288227600,0aa046,0aa091,0b111f8282886788766866852288586882288228216660606,0b2122,0b30438861,0b5042858868886668688228882078688228822760600660,0b805f28868686868686850633,0be09d,0c3127868681020222022882276066660,0c40de86788228728868886668688866606066860678688228886888681666868527285886872272882228166602,0c504b88688810600220222027686,0c60da86816,0c70a8,0c80d98527222027686,0c90b08886888666868822888886058228822820228287688868166768886816606006786667021,0cb046827222025888685272850,0cc11f8702288888605822882281660602,0d00d588886070220,0d60d98527222070070,0d811a86868850607,0d90b8786783,0da0268768822882876888666868886661600668666868886816611,0da107272858868886166786888616660066870202,0db0b586816,0dc02527201288227600,0dd0b48527222027686,0e301e8861,0e501d88888605822882222027686,0e50b0888860702,0eb0b48527222206702,0f401d88868886166702,0f40b1867888666868822888788228822202786,0f50298822882888860652288227600,0fc02d,1010b98765,10206f82222282220228888882828222006588606600660020066660660066868,10207288228888685866686888681636160706768166686882288878886706786882288868822885228166,10303e28216668688228887685886166702,1030b8220258527227,10502b87,10503a288282220065886066006,10602958882078688866683,10904e8527285886882288227600,10a0b127282887685272858122816,1160ad86885,11706327683,1170ab8216,11904f877,11905f87022872812220605886,119097882886888666868886661600658,11b04e858868886166783"; string[] k = split(s,","); for(string str:k) { string a = substr(str,0,3),b=substr(str,3,3),c=substr(str,6); real x=hex(a),y=hex(b); for (int i=0;i<length(c);++i) { int next = hex(substr(c,i,1)); real yI=(int)((next-next%3)/3),xI=next%3; --xI; --yI; draw((x,-y)--(x+xI,-(y+yI))); x+=xI; y+=yI; } }[/asy]

Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

Angle $A$ in triangle $ABC$ is equal to $a$. A circle passing through $A$ and $B$ and tangent to $BC$ intersects the median to side $BC$ (or its extension) at a point $M$ different from $A$. Find the angle $\angle BMC$.