Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.

Let $ABC,AA_1A_2,BB_1B_2, CC_1C_2$ be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments $A_2B_1,B_2C_1, C_2A_1$ by $P,Q,R$ in this order. Prove that the triangle $PQR$ is equilateral.

Let $ABCD$ be a cyclic quadrilateral and let $H_{A}, H_{B}, H_{C}, H_{D}$ be the orthocenters of the triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Show that the quadrilaterals $ABCD$ and $H_{A}H_{B}H_{C}H_{D}$ are congruent.

Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$. [i]Proposed by Patrik Bak, Slovakia[/i]

Let $ABCD$ be an unit aquare.$E,F$ be the midpoints of $CD,BC$ respectively. $AE$ intersects the diagonal $BD$ at $P$. $AF$ intersects $BD,BE$ at $Q,R$ respectively. Find the area of quadrilateral $PQRE$.

The word "'''HELP'''" in block letters is painted in black with strokes $1$ unit wide on a $5$ by $15$ rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is [asy] unitsize(12); fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black); fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black); fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black); fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black); fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white); draw((0,0)--(15,0)--(15,5)--(0,5)--cycle); label("$5\left\{ \begin{tabular}{c} \\ \\ \\ \\ \end{tabular}\right.$",(1,2.5),W); label(rotate(90)*"$\{$",(0.5,0.1),S); label("$1$",(0.5,-0.6),S); label(rotate(90)*"$\{$",(3.5,0.1),S); label("$1$",(3.5,-0.6),S); label(rotate(90)*"$\{$",(7.5,0.1),S); label("$1$",(7.5,-0.6),S); label(rotate(90)*"$\{$",(11.5,0.1),S); label("$1$",(11.5,-0.6),S); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(1.5,4),N); label("$3$",(1.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(5.5,4),N); label("$3$",(5.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(9.5,4),N); label("$3$",(9.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(13.5,4),N); label("$3$",(13.5,5.8),N); label("$\left. \begin{tabular}{c} \\ \end{tabular}\right\} 2$",(14,1),E); [/asy] $\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38$

Let $ABC$ an isosceles triangle, $P$ a point belonging to its interior. Denote $M$, $N$ the intersection points of the circle $\mathcal{C}(A, AP)$ with the sides $AB$ and $AC$, respectively. Find the position of $P$ if $MN+BP+CP$ is minimum.

Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$. [i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]

Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal. [i]Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.[/i]

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.

The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.

[b]p1.[/b] Four witches are riding their brooms around a circle with circumference $10$ m. They are standing at the same spot, and then they all start to ride clockwise with the speed of $1$, $2$, $3$, and $4$ m/s, respectively. Assume that they stop at the time when every pair of witches has met for at least two times (the first position before they start counts as one time). What is the total distance all the four witches have travelled? [b]p2.[/b] Suppose $A$ is an equilateral triangle, $O$ is its inscribed circle, and $B$ is another equilateral triangle inscribed in $O$. Denote the area of triangle $T$ as $[T]$. Evaluate $\frac{[A]}{[B]}$. [b]p3. [/b]Tim has bought a lot of candies for Halloween, but unfortunately, he forgot the exact number of candies he has. He only remembers that it's an even number less than $2020$. As Tim tries to put the candies into his unlimited supply of boxes, he finds that there will be $1$ candy left if he puts seven in each box, $6$ left if he puts eleven in each box, and $3$ left if he puts thirteen in each box. Given the above information, find the total number of candies Tim has bought. [b]p4.[/b] Let $f(n)$ be a function defined on positive integers n such that $f(1) = 0$, and $f(p) = 1$ for all prime numbers $p$, and $$f(mn) = nf(m) + mf(n)$$ for all positive integers $m$ and $n$. Let $$n = 277945762500 = 2^23^35^57^7$$ Compute the value of $\frac{f(n)}{n}$ . [b]p5.[/b] Compute the only positive integer value of $\frac{404}{r^2-4}$ , where $r$ is a rational number. [b]p6.[/b] Let $a = 3 +\sqrt{10}$ . If $$\prod^{\infty}_{k=1} \left( 1 + \frac{5a + 1}{a^k + a} \right)= m +\sqrt{n},$$ where $m$ and $n$ are integers, find $10m + n$. [b]p7.[/b] Charlie is watching a spider in the center of a hexagonal web of side length $4$. The web also consists of threads that form equilateral triangles of side length $1$ that perfectly tile the hexagon. Each minute, the spider moves unit distance along one thread. If $\frac{m}{n}$ is the probability, in lowest terms, that after four minutes the spider is either at the edge of her web or in the center, find the value of $m + n$. [b]p8.[/b] Let $ABC$ be a triangle with $AB = 10$; $AC = 12$, and $\omega$ its circumcircle. Let $F$ and $G$ be points on $\overline{AC}$ such that $AF = 2$, $FG = 6$, and $GC = 4$, and let $\overrightarrow{BF}$ and $\overrightarrow{BG}$ intersect $\omega$ at $D$ and $E$, respectively. Given that $AC$ and $DE$ are parallel, what is the square of the length of $BC$? [b]p9.[/b] Two blue devils and $4$ angels go trick-or-treating. They randomly split up into $3$ non-empty groups. Let $p$ be the probability that in at least one of these groups, the number of angels is nonzero and no more than the number of devils in that group. If $p = \frac{m}{n}$ in lowest terms, compute $m + n$. [b]p10.[/b] We know that$$2^{22000} = \underbrace{4569878...229376}_{6623\,\,\, digits}.$$ For how many positive integers $n < 22000$ is it also true that the first digit of $2^n$ is $4$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The lengths of the sides of triangle $A'B'C'$ are equal to the lengths of the three medians of triangle $ABC.$ Then the ratio $\mathrm{Area} (A'B'C') / \mathrm{Area} (ABC)$ equals $$ \mathrm a. ~ \frac 12\qquad \mathrm b.~\frac 23\qquad \mathrm c. ~\frac34 \qquad \mathrm d. ~\frac56 \qquad \mathrm e. ~\text{Cannot be determined from the information given.} $$

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.

Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.

A circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.

Let $ ABCDE$ be convex pentagon such that $ S(ABC) \equal{} S(BCD) \equal{} S(CDE) \equal{} S(DEA) \equal{} S(EAB)$. Prove that there is a point $ M$ inside pentagon such that $ S(MAB) \equal{} S(MBC) \equal{} S(MCD) \equal{} S(MDE) \equal{} S(MEA)$.

Consider triangle $ABC$ with orthocenter $H$. Let points $M$ and $N$ be the midpoints of segments $BC$ and $AH$. Point $D$ lies on line $MH$ so that $AD\parallel BC$ and point $K$ lies on line $AH$ so that $DNMK$ is cyclic. Points $E$ and $F$ lie on lines $AC$ and $AB$ such that $\angle EHM=\angle C$ and $\angle FHM=\angle B$. Prove that points $D,E,F$ and $K$ lie on a circle. [i]Proposed by Alireza Dadgarnia[/i]

If one side of a triangle is $ 12$ inches and the opposite angle is $ 30$ degrees, then the diameter of the circumscribed circle is: $ \textbf{(A)}\ 18\text{ inches} \qquad\textbf{(B)}\ 30\text{ inches} \qquad\textbf{(C)}\ 24\text{ inches} \qquad\textbf{(D)}\ 20\text{ inches}\\ \textbf{(E)}\ \text{none of these}$