Školjka

%V0 Let $ABCD$ be a convex quadrilateral. The diagonals $AC$ and $BD$ intersect at $K$. Show that $ABCD$ is cyclic if and only if $AK \sin A + CK \sin C = BK \sin B + DK \sin D$.

%V0 The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC \perp BD.$ Four squares, $ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $CL, DF, AH, BJ$ are denoted by $P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $AI, BK, CE DG$ are denoted by $P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $P_1,Q_1,R_1, S_1$ and $P_2,Q_2,R_2, S_2$ are the two quadrilaterals. Alternative formulation: Outside a convex quadrilateral $ABCD$ with perpendicular diagonals, four squares $AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $Q_1$ and $Q_2$ formed by the lines $AG, BI, CK, DE$ and $AJ, BL, CF, DH,$ respectively, are congruent.

%V0 Let $R$ be a rectangle that is the union of a finite number of rectangles $R_i,$ $1 \leq i \leq n,$ satisfying the following conditions: (i) The sides of every rectangle $R_i$ are parallel to the sides of $R.$ (ii) The interiors of any two different rectangles $R_i$ are disjoint. (iii) Each rectangle $R_i$ has at least one side of integral length. Prove that $R$ has at least one side of integral length. Variant: Same problem but with rectangular parallelepipeds having at least one integral side.

%V0 Show that any two points lying inside a regular $n-$gon $E$ can be joined by two circular arcs lying inside $E$ and meeting at an angle of at least $\left(1 - \frac{2}{n} \right) \cdot \pi.$

%V0 In the convex pentagon $ABCDE,$ the sides $BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($AC$ is parallel to $DE$, $BD$ is parallel to $AE$ etc.). Prove that $ABCDE$ is a regular pentagon.

%V0 Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.