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1Let \(T_1\) and \(T_2\) be the centroids of triangles \(A_1B_1C_1\) and \(A_2B_2C_2\), respectively. Prove that \[ \overrightarrow{A_1A_2} + \overrightarrow{B_1B_2} + \overrightarrow{C_1C_2} = 3\,\overrightarrow{T_1T_2}. \] …Open2Exactly \(2021\) points are chosen on the line \(AB\), and none of them lies on the segment \(AB\). Prove that the sum of the distances from these \(2021\) points to \(A\) can never be equal to the sum …3A triangle is cut into two triangles that are congruent to each other. Prove that the original triangle is isosceles.4In a triangle \(ABC\), the altitude from \(A\) meets the line \(BC\) at \(D\), and its length satisfies \[ AD = \tfrac{1}{2}\,BC. \] Prove that the angle of the triangle at the vertex \(A\) cannot be obtuse. …5A circle is inscribed in triangle \(ABC\), whose sides have lengths \(BC = a\), \(CA = b\) and \(AB = c\). A line tangent to this circle meets the side \(AC\) at the point \(P\) and the side \(BC\) at …6In a triangle \(ABC\), let \(C_1\) be the midpoint of the side \(AB\), so that \(CC_1\) is the median from \(C\). Let \(K\) be the midpoint of the segment \(CC_1\), and let the line \(AK\) meet the side …7Let \(X\) be a point in the interior of triangle \(ABC\), and let \(T\) be the centroid of that triangle. Points \(M\) and \(N\) lie on side \(BC\), points \(P\) and \(Q\) lie on side \(CA\), and points …8Let \(O\) and \(H\) be the circumcenter and the orthocenter of a triangle \(ABC\), and let \(G_1\), \(G_2\), \(G_3\) be the centroids of the triangles \(HBC\), \(HCA\), \(HAB\), respectively. Prove that …9Let \(ABCD\) be a square and let \(E\) be the midpoint of its side \(CD\). The line through \(D\) perpendicular to the diagonal \(BD\) meets the line \(AE\) at a point \(F\). Prove that the points \(B\), …10In a quadrilateral \(ABCD\), \[ \angle ABC = 104^\circ, \qquad \angle ADC = 128^\circ, \qquad AB = BC = 2. \] Compute the length of the diagonal \(BD\).11Let \(AA_0\), \(BB_0\) and \(CC_0\) be the altitudes of a triangle \(ABC\), and let \(H\) be its orthocenter. Let \(M\) be the midpoint of the segment \(AA_0\) and let \(N\) be the midpoint of the segment …12Does there exist a triangle of area \(1\) whose sides \(b\) and \(c\) satisfy \(c \le b\) and \(b = 1.4\)?13An angle of \(7^\circ\) is given. Using only compass and straightedge, divide it into seven equal parts.14In a trapezoid \(ABCD\) with \(AB \parallel CD\), the two angles at the base \(AB\) add up to \(90^\circ\). Prove that the segment joining the midpoints of the two bases has length equal to half the difference …15In triangle \(ABC\) the angle at \(B\) equals \(60^\circ\). The bisector of \(\angle CAB\) meets the opposite side at \(D\), the bisector of \(\angle BCA\) meets the opposite side at \(E\), and \(S\) is …16The quadrilateral \(ABCD\) is inscribed in a circle, and its diagonal \(AC\) is a diameter of that circle. Prove that the projections of the sides \(AB\) and \(CD\) onto the diagonal \(BD\) are equal.17Let \(K\) be the midpoint of the side \(CD\) of a rectangle \(ABCD\). The lines \(BK\) and \(AC\) are perpendicular to each other and meet at the point \(H\), and \(G\) denotes the foot of the perpendicular …18Prove that in the regular octagon \(A_1A_2A_3A_4A_5A_6A_7A_8\) the diagonals \(A_1A_6\), \(A_3A_7\) and \(A_5A_8\) pass through one point.19Three distinct points \(A\), \(B\), \(C\) lie on a line \(\ell\), and a point \(O\) lies off \(\ell\). The perpendicular bisectors of the segments \(OA\), \(OB\) and \(OC\) form a triangle \(EFG\). Prove …20Let \(xOy\) be an angle, and let \(A\), \(B\) and \(C\) be points on the arm \(Ox\) such that \(OA = 3\), \(OB = 4\) and \(OC = 6\). Let \(D\) be the foot of the perpendicular dropped from \(B\) to the …
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