The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$

The diagonals of a convex pentagon divide each of its interior angles into three equal parts. Does it follow that the pentagon is regular?

It is known that there is a straight line dividing the perimeter and area of a certain polygon circumscribed around a circle in the same ratio. Prove that this line passes through the center of the indicated circle.

Three line segments divide a triangle into five triangles. The area of these triangles is called $u, v, x,$ yand $z$, as in the figure. (a) Prove that $uv = yz$. (b) Prove that the area of the great triangle is at most $ \frac{xz}{y}$ [img]https://cdn.artofproblemsolving.com/attachments/9/4/2041d62d014cf742876e01dd8c604c4d38a167.png[/img]

A bee travels in a series of steps of length $1$: north, west, north, west, up, south, east, south, east, down. (The bee can move in three dimensions, so north is distinct from up.) There exists a plane $P$ that passes through the midpoints of each step. Suppose we orthogonally project the bee’s path onto the plane $P$, and let $A$ be the area of the resulting figure. What is $A^2$?

In triangle $ABC$ medians of triangle $BE$ and $AD$ are perpendicular to each other. Find the length of $\overline{AB}$, if $\overline{BC}=6$ and $\overline{AC}=8$

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$

We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$. a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN. b) The circle is not CN. Which one of these sets are CN? 1) $A=\{x\in\mathbb R^3| |x|=1\}$ 2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$ 3) Graph of the function $f:[0,1]\to \mathbb R$ defined by \[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

Let $ABCD$ be a parallelogram and a circle $k$ passes through $A, C$ and meets rays $AB, AD$ at $E, F$. If $BD, EF$ and the tangent at $C$ concur, show that $AC$ is diameter of $k$.

Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$. Let $A$ and $B$ be two points on the segment $XY$. The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$, respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$. Let $M$ the the intersection point between $XY$ and $NP$. Prove that $M$ is the midpoint of the segment $AB$.

The maximum number of solid regular tetrahedrons can be placed against each other so that one of their edges coincides with a given line segment in space? [hide=original wording]Hoeveel massieve regelmatige viervlakken kan men maximaal tegen mekaar plaatsen zodat ´e´en van hun ribben samenvalt met een gegeven lijnstuk in de ruimte?[/hide]

Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.

Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$. Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.

Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$. a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible. b) Find the smallest possible value of $CK+KL$. (Olexii Panasenko)

As shown in the figure, it is known that $BC= AC$ in $\vartriangle ABC$, $M$ is the midpoint of $AB$, points $D$, $E$ lie on $AB$ such that $\angle DCE= \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F $(different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$, $O_2$ respectively. Prove that $O_1O_2 \perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/8/c/62d4ecbc18458fb4f2bf88258d5024cddbc3b0.jpg[/img]

In the quadrilateral $ABCD$, the points $E, F$, and $K$ are midpoints of the $AB, BC, AD$ respectively. Known that $KE \perp AB, K F \perp BC$, and the angle $\angle ABC = 118^o$. Find $ \angle ACD$ (in degrees).

Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar. (A. Zaslavsky)

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals [asy] size(230); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(36), C=intersectionpoint(B--(9001,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(30)); label("$19$", (A+C)/2, dir(A--C)*dir(90)); label("$14$", (A+B)/2, dir(A--B)*dir(270)); [/asy] $\displaystyle \text{(A)} \ 2 \qquad \text{(B)} \ \frac{5}{2} \qquad \text{(C)} \ \frac{5}{2} - \sin \theta \qquad \text{(D)} \ \frac{5}{2} - \frac{1}{2} \sin \theta \qquad \text{(E)} \ \frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$

Let $ABC$ be an acute triangle with altitudes $AA'$ and $BB'$ and orthocenter $H$. Let $C_0$ be the midpoint of the segment $AB$. Let $g$ be the line symmetric to the line $CC_0$ with respect to the angular bisector of $\angle ACB$. Let $h$ be the line symmetric to the line $HC_0$ with respect to the angular bisector of $\angle AHB$. Show that the lines $g$ and $h$ intersect on the line $A'B'$.

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

A circle touches to diameter $AB$ of a unit circle with center $O$ at $T$ where $OT>1$. These circles intersect at two different points $C$ and $D$. The circle through $O$, $D$, and $C$ meet the line $AB$ at $P$ different from $O$. Show that \[|PA|\cdot |PB| = \dfrac {|PT|^2}{|OT|^2}.\]

Describe a construction of quadrilateral $ABCD$ given: (i) the lengths of all four sides; (ii) that $AB$ and $CD$ are parallel; (iii) that $BC$ and $DA$ do not intersect.

In the triangle $ABC$ we have $|AB|<|AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$. Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$