Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.

$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that \[\cos B+\cos C=1\Longleftrightarrow MI=MT\]

Let $ABCD$ be a parallelogram. Let $E$ and $F$ be the midpoints of sides $AB$ and $BC$ respectively. The lines $EC$ and $FD$ intersect at $P$ and form four triangles $APB, BPC, CPD, DPA$. If the area of the parallelogram is $100$, what is the maximum area of a triangles among these four triangles?

Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that \[ AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.\]

Given a parallelogram $ABCD$, in which the angle $\angle ABC$ is obtuse. Line $AD$ intersects the circle a second time $\omega$ circumscribed around triangle $ABC$, at the point $E$. Line $CD$ intersects second time circle $\omega$ at point $F$. Prove that the circumcenter of triangle $DEF$ lies on the circle $\omega$.

In the acute-angled triangle $ ABC $, the sides $ AB $ and $BC$ have different lengths, and the extension of the median $ BM $ intersects the circumscribed circle at the point $ N $. On this circle we note such a point $ D $ that $ \angle BDH = 90 {} ^ \circ $, where $ H $ is the point of intersection of the altitudes of the triangle $ ABC $. The point $K$ is chosen so that $ ANCK $ is a parallelogram. Prove that the lines $ AC $, $ KH $ and $ BD $ intersect at one point. (Igor Nagel)

Let the medians of the triangle $ABC$ meet at $G$. Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$, and let $P$ and $Q$ be points on the segments $BD$ and $BE$, respectively, such that $2BP=PD$ and $2BQ=QE$. Determine $\angle PGQ$.

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA \plus{} < MCB \equal{} MND \plus{} < MBC \equal{} 180^0$. Prove that $ MN$ is parallel to $ AB$.

Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$. Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.

Given triangle $ ABC$ with medians $ AE$, $ BF$, $ CD$; $ FH$ parallel and equal to $ AE$; $ BH$ and $ HE$ are drawn; $ FE$ extended meets $ BH$ in $ G$. Which one of the following statements is not necessarily correct? $ \textbf{(A)}\ AEHF \text{ is a parallelogram} \qquad \textbf{(B)}\ HE\equal{}HG \\ \textbf{(C)}\ BH\equal{}DC \qquad \textbf{(D)}\ FG\equal{}\frac{3}{4}AB \qquad \textbf{(E)}\ FG\text{ is a median of triangle }BFH$

Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$. Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that \[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

In a quadrilateral $ABCD$ the intersection of the diagonals is called $P$. Point $X$ is the orthocentre of triangle $PAB$. (The orthocentre of a triangle is the point where the three altitudes of the triangle intersect.) Point $Y$ is the orthocentre of triangle $PCD$. Suppose that $X$ lies inside triangle $PAB$ and $Y$ lies inside triangle $PCD$. Moreover, suppose that $P$ is the midpoint of line segment $XY$ . Prove that $ABCD$ is a parallelogram. [asy] import geometry; unitsize (1.5 cm); pair A, B, C, D, P, X, Y; A = (0,0); B = (2,-0.5); C = (3.5,2.2); D = A + C - B; P = (A + C)/2; X = orthocentercenter(A,B,P); Y = orthocentercenter(C,D,P); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(A--extension(A,X,B,P), dotted); draw(B--extension(B,X,A,P), dotted); draw(P--extension(P,X,A,B), dotted); draw(C--extension(C,Y,D,P), dotted); draw(D--extension(D,Y,C,P), dotted); draw(P--extension(P,Y,C,D), dotted); dot("$A$", A, W); dot("$B$", B, S); dot("$C$", C, E); dot("$D$", D, N); dot("$P$", P, E); dot("$X$", X, NW); dot("$Y$", Y, SE); [/asy]

In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.

A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.

5. Let $\triangle$ABC be a triangle with circumcentre [i]O[/i]. The perpendicular line from [i]O[/i] to [i]BC[/i] intersects line [i]BC[/i] at [i]M[/i] and line [i]AC[/i] at [i]P[/i], and the perpendicular line from [i]O[/i] to [i]AC[/i] intersects line [i]AC[/i] at [i]N[/i] and line [i]BC[/i] at [i]Q[/i]. Let [i]D[/i] be the intersection point of lines [i]PQ[/i] and [i]MN[/i]. construct the parallelogram [i]PCQJ[/i] with [i]PJ[/i] || [i]CQ[/i] and [i]QJ[/i] || [i]CP[/i]. Prove the following: a) The points [i]A[/i], [i]B[/i], [i]O[/i], [i]P[/i], [i]Q[/i], [i]J[/i] are all on the same circle. b) line [i]OD[/i] is perpendicular to line [i]CJ[/i].

Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.

Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .

Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$. (Greece)

Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.

Given a square $ABCD$, let $P, Q, R$, and $S$ be four variable points on the sides $AB, BC, CD$, and $DA$, respectively. Determine the positions of the points $P, Q, R$, and $S$ for which the quadrilateral $PQRS$ is a parallelogram, a rectangle, a square, or a trapezoid.