A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).

Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral. [i]Proposed by Owen Goff[/i]

Given isosceles $\triangle ABC$ with $AB = AC$ and $\angle BAC = 22^{\circ}$. On the side $BC$ point $D$ is chosen such that $BD = 2CD$. The foots of perpendiculars from $B$ to lines $AD$ and $AC$ are points $E$, $F$ respectively. Find with the proof value of the angle $\angle CEF$.

Four perpendiculars are drawn from four vertices of a convex pentagon to the opposite sides. If these four lines pass through the same point, prove that the perpendicular from the fifth vertex to the opposite side also passes through this point.

In the acute triangle $ABC$ point $I$ is the incenter, $O$ is the circumcenter, while $I_a$ is the excenter opposite the vertex $A$. Point $A'$ is the reflection of $A$ across the line $BC$. Prove that angles $\angle IOI_a$ and $\angle IA'I_a$ are equal.

Given a triangle $PQR$, the inscribed circle $\omega$ which touches the sides $QR, RP$ and $PQ$ at points $A, B$ and $C$, respectively, and $AB^2 + AC^2 = 2BC^2$. Prove that the point of intersection of the segments $PA, QB$ and $RC$, the center of the circle $\omega$, the point of intersection of the medians of the triangle $ABC$, the point $A$ and the midpoints of the segments $AC$ and $AB$ lie on one circle.

Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD\equal{}DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC\equal{}2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.

Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$.

Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.

Position the $4$ points on plane so that when measuring of all pairwise distances between them, it turned out only two different numbers. Find all such locations.

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].

A cylindrical container has height $6 cm$ and radius $4 cm$. It rests on a circular hoop, also of radius $4 cm$, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop in two points. Find with proof the locus of one of the cylinder’s vertical ends.

Six regular octagons each with sides of length 2 are placed in a two-by-three array and inscribed in a square as shown. The area of the square can be written in the form $m+n\sqrt{2}$ where $m$ and $n$ are positive integers. Find $m+n$. [asy] size(175); pair A=dir(-45/2); real h=A.x; path P=A; for(int k=1;k<8;++k) P=P--rotate(45*k)*A; P=P--cycle; for(real x=-2*h;x<3*h;x+=2*h) for(real y:new real[]{-h,h} ) draw (shift((x,y))*P); pair C=dir(45/2)+(2*h,h), X=(C.x+C.y,0), Y=(0,C.x+C.y); draw(X--Y--(-X)--(-Y)--cycle);[/asy]

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively. Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$

The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $C=90$) so that points $A_1$, $B_1$ lie on the legs $CB$ and $CA$ respectively ,and points $C_1$, $D_1$ lie on the hypotenuse $AB$. The circumcircle of triangles $B_1A_1C$ an $AC_1B_1$ intersect at $B_1$ and $Y$. Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let $X =\{ X_1 , X_2 , ...\}$ be a countable set of points in space. Show that there is a positive sequence $\{a_k\}$ such that for any point $Z\not\in X$ the distance between the point Z and the set $\{X_1,X_2 , ...,X_k\}$ is at least $a_k$ for infinitely many k.

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}\plus{} \tfrac{1}{PC}$ is a maximum. [asy] size(200); defaultpen(linewidth(0.7)); pair A = origin, B = (5,0), C = (4.2,3), P = waypoint(B--C,0.65); pair Bp = 1.3 * B, Cp = 1.2 * C; draw(A--B--C--A^^Bp--A--Cp); dot(P); label("$A$",A,W); label("$B$",B,S); label("$C$",C,dir(B--C)); label("$P$",P,dir(A--P)); [/asy]

Given a triangle $ABC$. Find the minimum of \[\frac{\cos^{2}\frac{A}{2}\cos^{2}\frac{B}{2}}{\cos^{2}\frac{C}{2}}+\frac{\cos^{2}\frac{B}{2}\cos^{2}\frac{C}{2}}{\cos^{2}\frac{A}{2}}+\frac{\cos^{2}\frac{C}{2}\cos^{2}\frac{A}{2}}{\cos^{2}\frac{B}{2}}. \]

In a convex hexagon $ABCDEF$ the diagonals $AD,BE,CF$ intersect at a point $M$. It is known that the triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are acute. It is also known that the quadrilaterals $ABDE,BCEF,CDFA$ have the same area. Prove that the circumcenters of triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are concyclic. [i]Proposed by Nairy Sedrakyan[/i]

Let $ABC$ be triangle with $A<B<C$ and inscribed in a circle $(O)$. On the minor arc $ABC$ of $(O)$ and does not contain point $A$, choose an arbitrary point $D$. Suppose $CD$ meets $AB$ at $E$ and $BD$ meets $AC$ at $F$. Let $O_1$ be the incenter of triangle $EBD$ touches with $EB,ED$ and tangent to $(O)$. Let $O_2$ be the incenter of triangle $FCD$, touches with $FC,FD$ and tangent to $(O)$. a) $M$ is a tangency point of $O_1$ with $BE$ and $N$ is a tangency point of $O_2$ with $CF$. Prove that the circle with diameter $MN$ has a fixed point. b) A line through $M$ is parallel to $CE$ meets $AC$ at $P$, a line through $N$ is parallel to $BF$ meets $AB$ at $Q$. Prove that the circumcircles of triangles $(AMP),(ANQ)$ are all tangent to a fixed circle.