The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.

Let $ABC$ an acute triangle with orthocenter $H$. Let $M$ be a point on segment $BC$. The line through $M$ and perpendicular to $BC$ intersects lines $BH$ and $CH$ in points $P$ and $Q$ respectively. Prove that the orthocenter of triangle $HPQ$ lies on the line $AM$.

$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of \[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]

The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that $(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$ and $(a-c)^2\leq 2b^2\leq (a+c)^2$.

In this diagram, not drawn to scale, figures $\text{I}$ and $\text{III}$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $\text{II}$ is a square region with area $32$ sq. in. Let the length of segment $AD$ be decreased by $12\frac{1}{2} \%$ of itself, while the lengths of $AB$ and $CD$ remain unchanged. The percent decrease in the area of the square is: [asy] draw((0,0)--(22.6,0)); draw((0,0)--(5.66,9.8)--(11.3,0)--(11.3,5.66)--(16.96,5.66)--(16.96,0)--(19.45,4.9)--(22.6,0)); label("A", (0,0), S); label("B", (11.3,0), S); label("C", (16.96,0), S); label("D", (22.6,0), S); label("I", (5.66, 3.9)); label("II", (14.15,2.83)); label("III", (19.7,2)); [/asy] $\textbf{(A)}\ 12\frac{1}{2} \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 87\frac{1}{2}$

The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.

The points $D, E, F$ lie respectively on the sides $BC$, $CA$, $AB$ of the triangle ABC such that $F B = BD$, $DC = CE$, and the lines $EF$ and $BC$ are parallel. Tangent to the circumscribed circle of triangle $DEF$ at point $F$ intersects line $AD$ at point $P$. Perpendicular bisector of segment $EF$ intersects the segment $AC$ at $Q$. Prove that the lines $P Q$ and $BC$ are parallel.

Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered: $\bullet$ those that contain the faces of the tetrahedron, and $\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above. Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then: (a) the maximum of $f(X)$ is equal to $9d^2+3+6d$; (b) the minimum of $f(X)$ is equal to $9d^2+3-6d$. [i]K. Dochev and I. Dimovski[/i]

Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle. [i]Andrew Gu[/i]

(A.Zaslavsky, 8) A triangle can be dissected into three equal triangles. Prove that some its angle is equal to $ 60^{\circ}$.

In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]

Prove that we can put in any arbitrary triangle with sidelengths $a,b,c$ such that $0\le a,b,c \le \sqrt2$ into a unit cube.

The circumference inscribed on the triangle $ABC$ is tangent to the sides $BC$, $CA$ and $AB$ on the points $D$, $E$ and $F$, respectively. $AD$ intersect the circumference on the point $Q$. Show that the line $EQ$ meet the segment $AF$ at its midpoint if and only if $AC=BC$.

[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person. (a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable. (b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable. [b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$? [b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$. (b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions. [b]p4.[/b] Let function $f(x, y, z)$ be defined as following: $$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$ Find the minimum value and prove the result. [b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.

Let $ABC$ be a triangle and let $L$, $M$, $N$ be the midpoints of the sides $BC$, $CA$ and $AB$ , respectively. The points $P$ and $Q$ lie on $AB$ and $BC$, respectively; the points $R$ and $S$ are such that $N$ is the midpoint of $PR$ and $L$ is the midpoint of $QS$. Show that if $PS$ and $QR$ are perpendicular, then their intersection lies on in the circumcircle of triangle $LMN$.

Angle bisector from vertex $A$ of acute triangle $ABC$ intersects side $BC$ in point $D$, and circumcircle of $ABC$ in point $E$ (different from $A$). Let $F$ and $G$ be foots of perpendiculars from point $D$ to sides $AB$ and $AC$. Prove that area of quadrilateral $AEFG$ is equal to the area of triangle $ABC$

$\triangle ABC$ is an acute triangle and its orthocenter is $H$. The circumcircle of $\triangle ABH$ intersects line $BC$ at $D$. Lines $DH$ and $AC$ meets at $P$, and the circumcenter of $\triangle ADP$ is $Q$. Prove that the circumcenter of $\triangle ABH$ lies on the circumcircle of $\triangle BDQ$.

Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$

Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$). [i]From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov[/i]

In quadrilateral $ABCD$, $CD = 14$, $\angle BAD = 105^o$, $\angle ACD = 35^o$, and $\angle ACB = 40^o$. Let the midpoint of $CD$ be $M$. Points $P$ and $Q$ lie on $\overrightarrow{AM}$ and $\overrightarrow{BM}$, respectively, such that $\angle AP B = 40^o$ and $\angle AQB = 40^o$ . $P B$ intersects $CD$ at point $R$ and $QA$ intersects $CD$ at point $S$. If $CR = 2$, what is the length of $SM$?

Let $ACE$ be a triangle with a point $B$ on segment $AC$ and a point $D$ on segment $CE$ such that $BD$ is parallel to $AE$. A point $Y$ is chosen on segment $AE$, and segment $CY$ is drawn. Let $X$ be the intersection of $CY$ and $BD$. If $CX = 5$, $XY = 3$, what is the ratio of the area of trapezoid $ABDE$ to the area of triangle $BCD$? [img]https://cdn.artofproblemsolving.com/attachments/9/2/d6c723e54dbc4c88c12aa6a6ee91ae9e3ea581.png[/img]

Let $ABC$ be a triangle with $AC>AB$ , and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC,CA,AB$ at $D,E,F$ respectively. Let $X$ and $Y$ be two points on minor arcs $\widehat{DF}$ and $\widehat{DE}$ of the incircle, respectively, such that $\angle BXD = \angle DYC$. Let line $XY$ meet line $BC$ at $K$. Let $T$ be the point on $\Omega$ such that $KT$ is tangent to $\Omega$ and $T$ is on the same side of line $BC$ as $A$. Prove that lines $TD$ and $AI$ meet on $\Omega$. [right][i]Tommy Walker Mackay, United Kingdom[/i][/right]

Prove that in the interior of an equilateral triangle with side $a$ you can put a finite number of equal circles that do not overlap, with radius $r = \frac{a}{2010}$, so that the sum of their areas is greater than $\frac{17\sqrt3}{80}$ a$^2$.