The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal. (M. Volchkevich)

Let $ABC$ be an acute-angled triangle in which $D,E,F$ are points on $BC,CA,AB$ respectively such that $AD \perp BC$;$AE = BC$; and $CF$ bisects $\angle C$ internally, Suppose $CF$ meets $AD$ and $DE$ in $M$ and $N$ respectively. If $FM$$= 2$, $MN =1$, $NC=3$, find the perimeter of $\Delta ABC$.

$ABCD $ is a cyclic quadrilateral such that the diagonals $AC $ and $BD $ are perpendicular and their intersection is $P $. Point $Q $ on the segment $CP$ is such that $CQ=AP $. Prove that the perimeter of triangle $BDQ $ is at least $2AC $. Tuymaada 2017 Q2 Juniors

Given a triangle $A_1 A_2 A_3$, let $a_i$ denote the side opposite to $A_i$, where indices are taken modulo 3. Let $D_1 \in a_1$. For $D_i \in A_i$, let $\omega_i$ be the incircle of the triangle formed by lines $a_i, a_{i+1}, A_iD_i$, and $D_{i+1} \in a_{i+1}$ with $A_{i+1} D_{i+1}$ tangent to $\omega_i$. Show that the set $\{D_i: i \in \mathbb{N}\}$ is finite.

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

Let $ ABCD $ be a rectangle of area $ S $, and $ P $ be a point inside it. We denote by $ a, b, c, d $ the distances from $ P $ to the vertices $ A, B, C, D $ respectively. Prove that $ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2\ge 2S $. When there is equality?

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 $BHD$.

Equilateral triangle $ T$ is inscribed in circle $ A$, which has radius $ 10$. Circle $ B$ with radius $ 3$ is internally tangent to circle $ A$ at one vertex of $ T$. Circles $ C$ and $ D$, both with radius $ 2$, are internally tangent to circle $ A$ at the other two vertices of $ T$. Circles $ B$, $ C$, and $ D$ are all externally tangent to circle $ E$, which has radius $ \frac {m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. [asy]unitsize(2.2mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep}; draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5)); dot(dotted); label("$E$",Ep,E); label("$A$",A,W); label("$B$",B,W); label("$C$",C,W); label("$D$",D,E);[/asy]

Take a point $O$ inside $\Delta A B C$ such that $\angle B O C=90^{\circ}$, $\angle C A O=\angle A B O$, $\angle B A O=\angle B C O .$ Find the value of $\frac{A C}{O C}$ [list=1] [*] $\sqrt{2}$ [*] $\sqrt{\frac{3}{2}}$ [*] 2 [*] None of these [/list]

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$, satisfying $AB<AC$. The tangent line at $A$ to the circumcicle of $ABC$ intersects $BC$ in $T$. Let $X$ be the midpoint of $AH$. Prove that $\angle ATX=\angle OTB$.

Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length $\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$ [size=70]*A polygon is called “simple” if it is not self intersecting.[/size]

Find all polygons $P$ that can be covered completely by three (possibly overlapping) smaller dilated versions of itself. [i]Proposed by Evan Chang (squareman), USA[/i]

p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$. Find the number of subsets of $H$. p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$. a. Give an example of your own from the above process. b. Explain mathematically the general form of the process. p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn? p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made. p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?

Let $ABC$ be a triangle, $H$ its orthocenter, and $\Gamma$ its circumcircle. Let $J$ be the point diametrically opposite to $A$ on $\Gamma$. The points $D$, $E$ and $F$ are the feet of the altitudes from $A$, $B$ and $C$, respectively. The line $AD$ intersects $\Gamma$ again at $P$. The circumcircle of $EFP$ intersects $\Gamma$ again at $Q$. Let $K$ be the second point of intersection of $JH$ with $\Gamma$. Prove that $K$, $D$ and $Q$ are collinear.

[b][b]p1.[/b][/b] Find the perimeter of a regular hexagon with apothem $3$. [b]p2.[/b] Concentric circles of radius $1$ and r are drawn on a circular dartboard of radius $5$. The probability that a randomly thrown dart lands between the two circles is $0.12$. Find $r$. [b]p3.[/b] Find all ordered pairs of integers $(x, y)$ with $0 \le x \le 100$, $0 \le y \le 100$ satisfying $$xy = (x - 22) (y + 15) .$$ [b]p4.[/b] Points $A_1$,$A_2$,$...$,$A_{12}$ are evenly spaced around a circle of radius $1$, but not necessarily in order. Given that chords $A_1A_2$, $A_3A_4$, and $A_5A_6$ have length $2$ and chords $A_7A_8$ and $A_9A_{10}$ have length $2 sin (\pi / 12)$, find all possible lengths for chord $A_{11}A_{12}$. [b]p5.[/b] Let $a$ be the number of digits of $2^{1998}$, and let $b$ be the number of digits in $5^{1998}$. Find $a + b$. [b]p6.[/b] Find the volume of the solid in $R^3$ defined by the equations $$x^2 + y^2 \le 2$$ $$x + y + |z| \le 3.$$ [b]p7.[/b] Positive integer $n$ is such that $3n$ has $28$ positive divisors and $4n$ has $36$ positive divisors. Find the number of positive divisors of $n$. [b]p8.[/b] Define functions $f$ and $g$ by $f (x) = x +\sqrt{x}$ and $g (x) = x + 1/4$. Compute $$g(f(g(f(g(f(g(f(3)))))))).$$ (Your answer must be in the form $a + b \sqrt{ c}$ where $a$, $b$, and $c$ are rational.) [b]p9.[/b] Sequence $(a_1, a_2,...)$ is defined recursively by $a_1 = 0$, $a_2 = 100$, and $a_n = 2a_{n-1}-a_{n-2}-3$. Find the greatest term in the sequence $(a_1, a_2,...)$. [b]p10.[/b] Points $X = (3/5, 0)$ and $Y = (0, 4/5)$ are located on a Cartesian coordinate system. Consider all line segments which (like $\overline{XY}$ ) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point $P$ on $\overline{XY}$ such that none of these line segments (except $\overline{XY}$ itself) pass through $P$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Segment equal to median $AA_0$ of triangle $ABC$ is drawn from point $A_0$ perpendicular to side $BC$ to the outer side of the triangle. Let's denote the second end of the constructed segment as $A_1$. Points $B_1$ and $C_1$ are constructed similarly. Find the angles of triangle $A_1B_1C_1$ if the angles of the triangle $ABC$ are $30^o$, $30^o$ and $120^o$. [hide=original wording]Медиану AA0 треугольника ABC отложили от точки A0 перпендикулярно стороне BC во внешнюю сторону треугольника. Обозначим второй конец построенного отрезка через A1. Аналогично строятся точки B1 и C1. Найдите углы треугольника A1B1C1, если углы треугольника ABC равны 30^o, 30^o и 120^o.[/hide]

Given a circle $\omega$, a point $A$ lying inside $\omega$, and point $B$ ($B \ne A$). All possible triangles $BXY$ are considered, such that the points $X$ and $Y$ lie on $\omega$ and the chord $XY$ passes through the point $A$. Prove that the centers of the circumcircles of the triangles $BXY$ lie on the same straight line.

[b]p1.[/b] There are $2000$ cans of paint. Show that at least one of the following two statements must be true. There are at least $45$ cans of the same color. There are at least $45$ cans all of different colors. [b]p2.[/b] The measures of the $3$ angles of one triangle are all different from each other but are the same as the measures of the $3$ angles of a second triangle. The lengths of $2$ sides of the first triangle are different from each other but are the same as the lengths of $2$ sides of the second triangle. Must the length of the remaining side of the first triangle be the same as the length of the remaining side of the second triangle? If yes, prove it. If not, provide an example. [b]p3.[/b] Consider the sequence $a_1=1$, $a_2=2$, $a_3=5/2$, ... satisfying $a_{n+1}=a_n+(a_n)^{-1}$ for $n>1$. Show that $a_{10000}>141$. [b]p4.[/b] Prove that no matter how $250$ points are placed in a disk of radius $1$, there is a disk of radius $1/10$ that contains at least $3$ of the points. [b]p5.[/b] Prove that: Given any $11$ integers (not necessarily distinct), one can select $6$ of them so that their sum is divisible by $6$. Given any $71$ integers (not necessarily distinct), one can select $36$ of them so that their sum is divisible by $36$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Circle $\omega$ with centre $M$ and diameter $XY$ is given. Point $A$ is chosen on $\omega$ so that $AX<AY$. Points $B, C$ are chosen on segments $XM, YM$, respectively, in a way that $BM=CM$. A parallel line to $AB$ is constructed through $C$; the line intersects $\omega$ at $P$ so that $P$ lies on the smaller arc $\widehat{AY}$. Similarly, a parallel line to $AC$ is constructed through $B$; the line intersects $\omega$ at $Q$ so that $Q$ lies on the smaller arc $\widehat{XA}$. Lines $PQ$ and $XY$ intersect at $S$. Prove that $AS$ is tangent to $\omega$.

Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that \[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]

Let $ABCD$ be a quadrilateral, with an inscribed circle with center $I$. Through $A$ are constructed perpendiculars to $AB$ and $AD$, which intersect $BI$ and $DI$ in points $M$ and $N$ respectively. Prove that $MN\perp AC$.

Let the excircle of a triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Denote by $\gamma$ the circumcircle of triangle $A_1B_1C_1$ and assume that $\gamma$ passes through vertex $A$. [list = a] [*] Show that $\overline{AA_1}$ is a diameter of $\gamma$. [*] Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$. [/list]

We consider an acute triangle $ABC$. The altitude from $A$ is $AD$, the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$. a) Prove that the triangles $ABC$ and $AF E$ are similar. b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.