Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.

Let $ABC$ be a triangle with $AB\ne AC$. Its incircle has center $I$ and touches the side $BC$ at point $D$. Line $AI$ intersects the circumcircle $\omega$ of triangle $ABC$ at $M$ and $DM$ intersects again $\omega$ at $P$. Prove that $\angle API= 90^o$.

An object has the shape of a square and has side length $a$. Light beams are shone on the object from a big machine. If $m$ is the mass of the object, $P$ is the power $\emph{per unit area}$ of the photons, $c$ is the speed of light, and $g$ is the acceleration of gravity, prove that the minimum value of $P$ such that the bar levitates due to the light beams is \[ P = \dfrac {4cmg}{5a^2}. \] [i]Problem proposed by Trung Phan[/i]

[b]p1.[/b] Find the minimum value of $x^4 +2x^3 +3x^2 +2x+2$, where x can be any real number. [b]p2.[/b] A type of digit-lock has $5$ digits, each digit chosen from $\{1,2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s? [b]p3.[/b] Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have $80\%$ chance to be good balls and MIO balls have $50\%$ chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball? [b]p4.[/b] What is the smallest positive integer that is one-ninth of its reverse? [b]p5.[/b] When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many dierent orders can Michael get dressed? [b]p6.[/b] The numbers $1, 2,..., 2015$ are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number? [b]p7.[/b] Let $A = (a_1b_1a_2b_2... a_nb_n)_{34}$ and $B = (b_1b_2... b_n)_{34}$ be two numbers written in base $34$. If the sum of the base-$34$ digits of $A$ is congruent to $15$ (mod $77$) and the sum of the base $34$ digits of $B$ is congruent to $23$ (mod $77$). Then if $(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x$ (mod $77$) and $0 \le x \le 76$, what is $x$? (you can write $x$ in base $10$) [b]p8.[/b] What is the sum of the medians of all nonempty subsets of $\{1, 2,..., 9\}$? [b]p9.[/b] Tony is moving on a straight line for $6$ minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes? [b]p10.[/b] Find the number of nonnegative integer triplets $a, b, c$ that satisfy $$2^a3^b + 9 = c^2.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three cities that are located on the vertices of an equilateral triangle with side length $100$ units. A missile flies in a straight line in the same plane as the equilateral triangle formed by the three citiies. The radar from City $A$ reported that the closest approach of the missile was $20$ units. The radar from City $B$ reported that the closest approach of the missile was $60$ units. However, the radar for city $C$ malfunctioned and did not report a distance. Find the minimum possible distance for the closest approach of the missile to city $C$.

In a triangle $ABC$ let $D,E,Z,H,G$ be the midpoints of $BC,AD,BD,ED,EZ$ respectively. Let $I$ be the intersection of $BE,AC$ and let $K$ be the intersection of $HG,AC$. Prove that: a) $AK=3CK$ b) $HK=3HG$ c) $BE=3EI$ d) $(EGH)=\frac{1}{32}(ABC)$ Notation $(...)$ stands for area of $....$

Through the circumcenter $O$ of an arbitrary acute-angled triangle, chords $A_1A_2,B_1B_2, C_1C_2$ are drawn parallel to the sides $BC,CA,AB$ of the triangle respectively. If $R$ is the radius of the circumcircle, prove that \[A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.\]

$$Problem 2 :$$If ABCD is as convex quadrilateral with $\angle ADC = 30$ and $BD = AB+BC+CA$, prove that $BD$ bisects $\angle ABC$.

Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$. For each triangle $T\in\Delta$, $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle. (a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$, among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$) at least two are equal. (b) Find the smallest $n$ with the property from (a).

Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that: \[PX || AC \ , \ PY ||AB \] Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$

Let $ABC$ be an acute triangle with circumcenter $O$, on the sides $BC, CA$ and $AB$ they take the points $D, E$ and $F$, respectively, in such a way that $BDEF$ is a parallelogram. Supposing that $DF^2 = AE\cdot EC <\frac{AC^2}{4}$ show that the circles circumscribed to the triangles $FBD$ and $AOC$ are tangent.

A triangle $ABC$ is given. The point $K$ is the base of the external bisector of angle $A$. The point $M$ is the midpoint of the arc $AC$ of the circumcircle. The point $N$ on the bisector of angle $C$ is such that $AN \parallel BM$. Prove that the points $M,N,K$ are collinear. [i](Proposed by Ilya Bogdanov)[/i]

[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$. [b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive? [b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$. [b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$. [b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ . [b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list? [b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$. [b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits? [b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces $$2\_3\_4\_5$$ [b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ . [b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes? p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence? [b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits. [b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$? [b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$. [b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$? [b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$ [b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$. [b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$. [b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number. [b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$? [b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers. [b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$. [b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute $$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$ [b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

It's given a right-angled triangle $ABC (\angle{C}=90^{\circ})$ and area $S$. Let $S_1$ be the area of the circle with diameter $AB$ and $k=\frac{S_1}{S}$\\ a) Compute the angles of $ABC$, if $k=2\pi$ b) Prove it is not possible for k to be $3$

In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.

In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.

(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.

A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 36$

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

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 isosceles triangle with $AB = AC$. Let $AD$ be the diameter of the circumcircle of $ABC$ and let $P$ be a point on the smaller arc $BD$. The line $DP$ intersects the rays $AB$ and $AC$ at points $M$ and $N$, respectively. The line $AD$ intersects the lines $BP$ and $CP$ at points $Q$ and $R$, respectively. Prove that the midpoint of $MN$ lies on the circumcircle of $PQR$

Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

Let \( ABC \) be an acute-angled scalene triangle. Let \( B_1 \) and \( B_2 \) be points on the rays \( BC \) and \( BA \), respectively, such that \( BB_1 = BB_2 = AC \). Similarly, let \( C_1 \) and \( C_2 \) be points on the rays \( CB \) and \( CA \), respectively, such that \( CC_1 = CC_2 = AB \). Prove that if \( B_1B_2 \) and \( C_1C_2 \) intersect at \( K \), then \( AK \) is parallel to \( BC \).

In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic. [i]Tiger Zhang[/i]