Let $n$ be a positive integer. Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$.

Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that (a) $M$ is the midpoint of $AB$; (b) $N$ is the midpoint of $AC$; (c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$. Prove that $\angle APM = \angle PBA$.

Two people A and B play a game in the $m \times n$ grid ($m,n \in \mathbb{N^*}$). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?

Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $

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 $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]

An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?

Let $ A_1A_2$ be the external tangent line to the nonintersecting cirlces $ \omega_1(O_1)$ and $ \omega_2(O_2)$,$ A_1\in\omega_1$,$ A_2\in\omega_2$.Points $ K$ is the midpoint of $ A_1A_2$.And $ KB_1$ and $ KB_2$ are tangent lines to $ \omega_1$ and $ \omega_2$,respectvely($ B_1\neq A_1$,$ B_2\neq A_2$).Lines $ A_1B_1$ and $ A_2B_2$ meet in point $ L$,and lines $ KL$ and $ O_1O_2$ meet in point $ P$. Prove that points $ B_1,B_2,P$ and $ L$ are concyclic.

The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is: $ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$ $ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$

A square of side length $ 1$ and a circle of radius $ \sqrt3/3$ share the same center. What is the area inside the circle, but outside the square? $ \textbf{(A)}\ \frac{\pi}3 \minus{} 1 \qquad\textbf{(B)}\ \frac{2\pi}{9} \minus{} \frac{\sqrt3}3 \qquad\textbf{(C)}\ \frac{\pi}{18} \qquad\textbf{(D)}\ \frac14 \qquad\textbf{(E)}\ 2\pi/9$

Find all infinite sequences $a_1, a_2, \ldots$ of positive integers satisfying the following properties: (a) $a_1 < a_2 < a_3 < \cdots$, (b) there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$, (c) there are infinitely many $k$ such that $a_k = 2k-1$.

Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common? $\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $

Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length? $ \textbf{(A) }\frac25\qquad\textbf{(B) }\frac49\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac59\qquad\textbf{(E) }\frac45 $

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.

Let $ABC$ be an acute triangle with $AB<AC<BC$, inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Let $O_1$ be the symmetric point of $O$ wrt $AC$. Circle $c_1(O_1,R)$ intersects $BC$ at $Z$. If the extension of the altitude $AD$ intersects the cicrumscribed circle $c(O,R)$ at point $E$, prove that $EC$ is perpendicular on $AZ$.

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$. [i]Bulgaria[/i]

In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game? $(\textbf{A})\: \frac{61}{216}\qquad(\textbf{B}) \: \frac{367}{1296}\qquad(\textbf{C}) \: \frac{41}{144}\qquad(\textbf{D}) \: \frac{185}{648}\qquad(\textbf{E}) \: \frac{11}{36}$

Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality \[ \frac {1}{a^3 \plus{} b^3 \plus{} abc} \plus{} \frac {1}{b^3 \plus{} c^3 \plus{} abc} \plus{} \frac {1}{c^3 \plus{} a^3 \plus{} abc} \leq \frac {1}{abc} \] holds.

A regular $15$-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$? $\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }32\qquad\textbf{(D) }39\qquad\textbf{(E) }54$

Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.

Let $AA', BB'$ and $CC'$ be the altitudes of an acute-angled triangle $ABC$. Points $C_a, C_b$ are symmetric to $C' $ wrt $AA'$ and $BB'$. Points $A_b, A_c, B_c, B_a$ are defined similarly. Prove that lines $A_bB_a, B_cC_b$ and $C_aA_c$ are parallel.

The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$