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1A circle can be inscribed in the trapezoid \(ABCD\), whose parallel sides are \(AB\) and \(CD\). Prove that the circle having \(BC\) as a diameter and the circle having \(AD\) as a diameter touch each …Open2Let \(ABC\) be an acute triangle and let \(D\) be the foot of the altitude from \(A\), so that \(D\) lies on the side \(BC\). A point \(P\) is chosen on the segment \(AD\) in such a way that \[ \angle PBA = \angle PCA . \] …3Let \(k > 0\). On the sides \(A_1B_1\), \(B_1C_1\) and \(C_1A_1\) of a triangle \(A_1B_1C_1\), points \(C_2\), \(A_2\) and \(B_2\) are chosen, respectively, so that \[ \frac{A_1C_2}{C_2B_1} = \frac{B_1A_2}{A_2C_1} = \frac{C_1B_2}{B_2A_1} = k . \] …4Let \(a\), \(b\), \(c\) be the side lengths of a triangle \(ABC\), let \(S\) be its area and \(R\) the radius of its circumscribed circle, and let \(M\) be a point in the interior of the triangle. Write …5In a triangle \(ABC\) we have \(\angle ABC = 45^\circ\) and \(\angle CAB = 15^\circ\). Let \(M\) be the point of the ray \(BC\) for which \[ \overrightarrow{BM} = 3 \cdot \overrightarrow{BC} . \] Determine …6Let \(ABCDEF\) be a regular hexagon. A point \(M\) is taken on the diagonal \(AC\) and a point \(N\) on the diagonal \(CE\) so that both are placed at the same relative position: \[ \frac{AM}{AC} = \frac{CN}{CE} = \lambda . \] …7Let \(ABC\) be an acute triangle with \(AB < AC\), and let \(D\) be the midpoint of its side \(BC\). Let \(p\) be the image of the line \(AD\) under reflection in the bisector of the angle \(BAC\), and …8Let \(ABC\) be a triangle with \(\angle CAB = 60^\circ\). Denote by \(O\) and \(I\) the centres of the circle circumscribed about it and of the circle inscribed in it, respectively, and let \(A'\) be the …9The figure below consists of a triangle \(ABC\) together with the circular segment erected on the side \(BC\), on the opposite side of \(BC\) from \(A\). Construct at least one straight line that divides …10Let \(H\) and \(O\) be the orthocentre and the circumcentre of a triangle \(ABC\) with \(AB \neq AC\). The lines \(AH\) and \(AO\) meet the circumcircle of \(ABC\) a second time at \(M\) and \(N\) respectively. …11Three points \(A\), \(B\), \(C\), not lying on one line, are given. Construct a point \(D\) for which the quadrilateral \(ABCD\) is at the same time cyclic and tangential, that is, admits both a circumscribed …12In a triangle \(ABC\) the angle at \(B\) equals \(80^\circ\). Three further points are marked: the point \(D\) on the side \(BC\) with \(AB = AD = CD\); the point \(F\) on the side \(AB\) with \(AF = BD\); …13Let \(ABC\) be an isosceles triangle with \(AB = AC\). A point \(P\) is taken inside the triangle so that \[ \angle BPC = 90^\circ + \tfrac{1}{2}\angle BAC , \] and a point \(Q\) is taken so that \(\angle BPQ = \angle PQA = 90^\circ\). …14Let \(ABC\) be a triangle. Prove that the following three lines all pass through one point: the bisector of the angle at \(A\); the line through the midpoints of the sides \(CA\) and \(CB\); and the line …15Let \(n\) be a natural number divisible by \(6\). Prove that there exists a convex \(n\)-gon whose interior angles are all equal and which can be cut into finitely many pieces, each piece being of one …16Let \(ABC\) be a triangle and let \(k\) be its circumcircle, with centre \(O\). Construct a point \(D\) on \(k\) such that the centroids of the triangles \(ABC\) and \(ABD\) are collinear with the point …17Let \(C\) be a point of the segment \(AB\) other than \(A\) and \(B\), and let \(k_{0}, k_{01}, k_{02}\) be the circles with diameters \(AB, AC\) and \(CB\) respectively. Let \(D\) be a point in which …18Let \(ABC\) be a right triangle with hypotenuse \(AB\), and let \(D\) be the midpoint of \(AB\). Let \(k\) be the circumcircle of the triangle \(BCD\), and let \(E\) be an arbitrary point of the shorter …19In a triangle \(ABC\), let \(O\) and \(I\) be the centres of the circumscribed and of the inscribed circle, respectively. Let \(O_1\) be the image of \(O\) under the reflection in the point \(I\), so that …20Determine the largest possible value of \(n\) for which there exists a convex \(n\)-gon that can be decomposed into a disjoint union of triangles, each of which is either right isosceles or right-angled …21Let \(ABC\) be a triangle, let \(D\) be the foot of the altitude from \(A\) to the line \(BC\), and let \(\omega\) be the circle whose diameter is the segment \(AD\). Denote by \(G\) the second common …22Circumscribe about a given triangle \(ABC\) an equilateral triangle \(PQR\) whose side is as long as possible. (Here \(\triangle PQR\) is called circumscribed about \(\triangle ABC\) when \(A \in QR\), …
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