Let $M$ be the midpoint of side $AB$ of the triangle $ABC$. Let$P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of the triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then $\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\ \text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\ \text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\ \text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\ \text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P$

Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?

Let O be the center of the circumscribed circle of the triangle ABC. Let AH be the altitude in this triangle, and let P be the base of the perpendicular drawn from point A to the line CO. Prove that the line HP passes through the midpoint of the side AB. (6 points) Egor Bakaev

Consider an inscriptible polygon $ABCDE$. Let $H_1,H_2,H_3,H_4,H_5$ be the orthocenters of the triangles $ABC,BCD,CDE,DEA,EAB$ and let $M_1,M_2,M_3,M_4,M_5$ be the midpoints of $DE,EA,AB,BC$ and $CD$, respectively. Prove that the lines $H_1M_1,H_2M_2,H_3M_3,H_4M_4,H_5M_5$ have a common point. [i]Dinu Serbanescu[/i]

Let $\Omega$ and $O$ be the circumcircle and the circumcenter of an acute triangle $ABC$ $(\overline{AB} < \overline{AC})$. Define $D,E(\neq A)$ be the points such that ray $AO$ intersects $BC$ and $\Omega$. Let the line passing through $D$ and perpendicular to $AB$ intersects $AC$ at $P$ and define $Q$ similarly. Tangents to $\Omega$ on $A,E$ intersects $BC$ at $X,Y$. Prove that $X,Y,P,Q$ lie on a circle.

Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' \equal{} BB' \equal{} CC' \equal{} x$. Prove that: a) If $ ABC\sim A'B'C'$ then $ x \equal{} 2r$ b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x \equal{} R \plus{} d$ or $ x \equal{} R \minus{} d$. (In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d \equal{} OI$)

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on the base $BC$ such that $BD:DC = 2: 1$. Note on the segment $AD$ a point $P$ such that $\angle BAC= \angle BPD $. Prove that $\angle BPD = 2 \angle CPD$.

Consider a sequence of eleven squares that have side lengths $3, 6, 9, 12,\ldots, 33$. Eleven copies of a single square each with area $A$ have the same total area as the total area of the eleven squares of the sequence. Find $A$.

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.

Let $A, B, C$ be points on a unit circle such that $|AB|=|BC|$ and $m(\widehat{ABC})=72^\circ$. Let $D$ be a point such that $\triangle BCD$ is equilateral. If $AD$ meets the circle at $D$, what is $|DE|$? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(C)}\ \dfrac {\sqrt 2}2 \qquad\textbf{(D)}\ \sqrt 3 -1 \qquad\textbf{(E)}\ \text{None of the above} $

Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.

Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\angle B = 90$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines $AH_c$ and $CH_a$ meet at point $K$. Prove that $\angle MBK = 90$.

In the Cartesian plane, a line segment with midpoint $(2020,11)$ has one endpoint at $(a,0)$ and the other endpoint on the line $y=x$. Compute $a$.

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 $AB$.

Given a circle $(O)$ with center $O$ and $P$ a point outside $(O)$. $A$ and $B$ are points on $(O)$ such that $PA$ and $PB$ are tangents to $(O)$. The line $\ell$ through $P$ intersects $(O)$ at points $C$ and $D$, respectively ($C$ lies between $P$ and $D$). Line $BF$ is parallel to line $PA$ and intersects line $AC$ and line $AD$ at $E$ and $F$, respectively. Prove that $BE = BF$.

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.

In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.

[u]Round 5[/u] [b]p13.[/b] Vincent the Bug is at the vertex $A$ of square $ABCD$. Each second, he moves to an adjacent vertex with equal probability. The probability that Vincent is again on vertex $A$ after $4$ seconds is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p14.[/b] Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, and $\angle BAC = 60^o$. Let $P$ be a point inside the triangle such that $BP = 1$ and $CP =\sqrt3$, let $x$ equal the area of $APC$. Compute $16x^2$. [b]p15.[/b] Let $n$ be the number of multiples of$ 3$ between $2^{2020}$ and $2^{2021}$. When $n$ is written in base two, how many digits in this representation are $1$? [u]Round 6[/u] [b]p16.[/b] Let $f(n)$ be the least positive integer with exactly n positive integer divisors. Find $\frac{f(200)}{f(50)}$ . [b]p17.[/b] The five points $A, B, C, D$, and $E$ lie in a plane. Vincent the Bug starts at point $A$ and, each minute, chooses a different point uniformly at random and crawls to it. Then the probability that Vincent is back at $A$ after $5$ minutes can be expressed as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p18.[/b] A circle is divided in the following way. First, four evenly spaced points $A, B, C, D$ are marked on its perimeter. Point $P$ is chosen inside the circle and the circle is cut along the rays $PA$, $PB$, $PC$, $PD$ into four pieces. The piece bounded by $PA$, $PB$, and minor arc $AB$ of the circle has area equal to one fifth of the area of the circle, and the piece bounded by $PB$, $PC$, and minor arc $BC$ has area equal to one third of the area of the circle. Suppose that the ratio between the area of the second largest piece and the area of the circle is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [u]Round 7 [/u] [b]p19.[/b] There exists an integer $n$ such that $|2^n - 5^{50}|$ is minimized. Compute $n$. [b]p20.[/b] For nonnegative integers $a = \overline{a_na_{n-1} ... a_2a_1}$, $b = \overline{b_mb_{m-1} ... b_2b_1}$, define their distance to be $$d(a, b) = \overline{|a_{\max\,\,(m,n)} - b_{\max\,\,(m,n)}||a_{\max\,\,(m,n)-1} - b_{\max\,\,(m,n)-1}|...|a_1 - b_1|}$$ where $a_k = 0$ if $k > n$, $b_k = 0$ if $k > m$. For example, $d(12321, 5067) = 13346$. For how many nonnegative integers $n$ is $d(2021, n) + d(12345, n)$ minimized? [b]p21.[/b] Let $ABCDE$ be a regular pentagon and let $P$ be a point outside the pentagon such that $\angle PEA = 6^o$ and $\angle PDC = 78^o$. Find the degree-measure of $\angle PBD$. [u]Round 8[/u] [b]p22.[/b] What is the least positive integer $n$ such that $\sqrt{n + 3} -\sqrt{n} < 0.02$ ? [b]p23.[/b] What is the greatest prime divisor of $20^4 + 21 \cdot 23 - 6$? [b]p24.[/b] Let $ABCD$ be a parallelogram and let $M$ be the midpoint of $AC$. Suppose the circumcircle of triangle $ABM$ intersects $BC$ again at $E$. Given that $AB = 5\sqrt2$, $AM = 5$, $\angle BAC$ is acute, and the area of $ABCD$ is $70$, what is the length of $DE$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949414p26408213]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a triangle $ABC$ with $\angle ACB = 120^{\circ}.$ A point $L$ is marked in the side $AB$ such that $CL$ bisects $\angle ACB.$ Points $N$ and $K$ are chosen in the sides $AC$ and $BC $ such that $CK+CN=CL.$ Prove that the triangle $KLN$ is equilateral.

A dart is thrown at a square dartboard of side length $2$ so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance $1$ of a corner?

Polar coordinate equation of curve $C$ is $\rho=1+\cos\theta$. Polar coordinate of point $A$ is $(2,0)$. $C$ rotate around $A$ for a whole circle, the area of the figure that $C$ swept out by is________.

Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

Each pair of three circles have the common chords $AA_1, BB_1$ and $CC_1{}$ such that lines $AB{}$ and $A_1B_1$ intersect in point $M{}$, $BC$ and $B_1C_1$ intersect in point $N{}$, $CA{}$ and $C_1A_1$ intersect in point $P{}$. Prove that points $M, N$ and $P$ are collinear.