Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$. [i]Proposed by Yang Liu[/i]

Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.

Draw the smaller number of line segments connecting points of the figure such that the new figure obtained to have exactly: [img]https://cdn.artofproblemsolving.com/attachments/d/1/098e03714904573a1eacd2d3dc28b4e8c42c7c.png[/img] i) one axis of symmetry ii) two axes of symmetry iii) four axes of symmetry Draw a new figure, at each case.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

[u]Round 1 [/u] [b]p1.[/b] Ben the bear has an algorithm he runs on positive integers- each second, if the integer is even, he divides it by $2$, and if the integer is odd, he adds $1$. The algorithm terminates after he reaches $1$. What is the least positive integer n such that Ben's algorithm performed on n will terminate after seven seconds? (For example, if Ben performed his algorithm on $3$, the algorithm would terminate after $3$ seconds: $3 \to 4 \to 2 \to 1$.) [b]p2.[/b] Suppose that a rectangle $R$ has length $p$ and width $q$, for prime integers $p$ and $q$. Rectangle $S$ has length $p + 1$ and width $q + 1$. The absolute difference in area between $S$ and $R$ is $21$. Find the sum of all possible values of $p$. [b]p3.[/b] Owen the origamian takes a rectangular $12 \times 16$ sheet of paper and folds it in half, along the diagonal, to form a shape. Find the area of this shape. [u]Round 2[/u] [b]p4.[/b] How many subsets of the set $\{G, O, Y, A, L, E\}$ contain the same number of consonants as vowels? (Assume that $Y$ is a consonant and not a vowel.) [b]p5.[/b] Suppose that trapezoid $ABCD$ satisfies $AB = BC = 5$, $CD = 12$, and $\angle ABC = \angle BCD = 90^o$. Let $AC$ and $BD$ intersect at $E$. The area of triangle $BEC$ can be expressed as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$. Find $a + b$. [b]p6.[/b] Find the largest integer $n$ for which $\frac{101^n + 103^n}{101^{n-1} + 103^{n-1}}$ is an integer. [u]Round 3[/u] [b]p7.[/b] For each positive integer n between $1$ and $1000$ (inclusive), Ben writes down a list of $n$'s factors, and then computes the median of that list. He notices that for some $n$, that median is actually a factor of $n$. Find the largest $n$ for which this is true. [b]p8.[/b] ([color=#f00]voided[/color]) Suppose triangle $ABC$ has $AB = 9$, $BC = 10$, and $CA = 17$. Let $x$ be the maximal possible area of a rectangle inscribed in $ABC$, such that two of its vertices lie on one side and the other two vertices lie on the other two sides, respectively. There exist three rectangles $R_1$, $R_2$, and $R_3$ such that each has an area of $x$. Find the area of the smallest region containing the set of points that lie in at least two of the rectangles $R_1$, $R_2$, and $R_3$. [b]p9.[/b] Let $a, b,$ and $c$ be the three smallest distinct positive values of $\theta$ satisfying $$\cos \theta + \cos 3\theta + ... + \cos 2021\theta = \sin \theta+ \sin 3 \theta+ ... + \sin 2021\theta. $$ What is $\frac{4044}{\pi}(a + b + c)$? [color=#f00]Problem 8 is voided. [/color] PS. You should use hide for answers.Rounds 4-5 have been posted [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $\angle A=60^{\circ}$, $AD$ be its bisector, and $PDQ$ be a regular triangle with altitude $DA$. The lines $PB$ and $QC$ meet at point $K$. Prove that $AK$ is a symmedian of $ABC$.

Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the following assertions: a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$. b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.

In triangle $ABC$, the incenter is $I$ and the circumcenter is $O$. Let $AI$ intersects $(ABC)$ second time at $P$ . The line passes through $I$ and perpendicular to $AI$ intersects $BC$ at $X$. The feet of the perpendicular from $X$ to $IO$ is $Y$. Prove that $A,P,X,Y$ cyclic.

Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

Daniel painted a rectangular painting measuring $2$ meters by $4$ meters with four colors. Knowing that he used more than two colors to paint the four corners of the painting, prove that he painted of the same color two points that are at least $\sqrt5$ meters

The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.

In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$. [i](Proposed by Danylo Khilko)[/i]

Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.

Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that \[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]

Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.

Let $ABC$ be an acute triangle. Let$H$ and $D$ be points on $[AC]$ and $[BC]$, respectively, such that $BH \perp AC$ and $HD \perp BC$. Let $O_1$ be the circumcenter of $\triangle ABH$, and $O_2$ be the circumcenter of $\triangle BHD$, and $O_3$ be the circumcenter of $\triangle HDC$. Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$.

An acute triangle ABC is inscribed in a circle C. Tangents in A and C to circle C intersect at F. Segment bisector of AB intersects the line BC at E. Show that the lines FE and AB are parallel.

In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles $< 1/2$.

Let $ ABC$ be an equilateral triangle of center $ O$, and $ M\in BC$. Let $ K,L$ be projections of $ M$ onto the sides $ AB$ and $ AC$ respectively. Prove that line $ OM$ passes through the midpoint of the segment $ KL$.

In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$. Prove that the line $PQ$ passes through the circumcenter of the triangle.

A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).

On square $ABCD$, point $E$ lies on side $\overline{AD}$ and point $F$ lies on side $\overline{BC}$, so that $BE=EF=FD=30$. Find the area of square $ABCD$.