Practice library
Problems
1Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.Open2The lengths of the sides of a triangle are three consecutive natural numbers, each greater than \(3\). The altitude drawn to the middle side splits that side into two segments. Prove that the lengths of …3Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.4For every point of the first quadrant, determine the line through that point which, together with the positive parts of the coordinate axes, bounds a triangle of the smallest possible area.5Two segments of lengths \(a\) and \(b\) are given. Construct a triangle \(ABC\) having these two segments as sides, in such a way that the angle opposite one of them is three times as large as the angle …6In a quadrilateral \(ABCD\) the sides \(AD\) and \(BC\) are equal, and the interior angles at \(A\) and \(B\) satisfy \[ \angle DAB + \angle ABC = 120^\circ . \] Prove that the midpoint of the diagonal …7The side lengths of a certain triangle are mutually distinct natural numbers, and its area is a natural number as well. Must that triangle be right-angled?8On the sides \(AB\) and \(BC\) of an equilateral triangle \(ABC\), points \(Z\) and \(X\) are chosen so that \[ AZ : ZB = BX : XC = 2021 : 2020. \] The perpendicular bisector of the segment \(XZ\) meets …9At most how many interior angles of a polygon with \(n\) vertices can be greater than \(180^\circ\)? (The polygon is simple: its sides meet only at the shared endpoints of neighbouring sides.) A \(n - 1\) …10Janez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …11A regular octagon is inscribed in a square of side length \(a\) so that four sides of the octagon lie on the four sides of the square. Express the side length of the octagon in terms of \(a\).12In triangle \(ABC\) the bisector of the angle \(CAB\) meets the side \(BC\) at the point \(N\), and the bisector of the angle \(CBA\) meets the side \(AC\) at the point \(P\), where \[ PN = a . \] Let …13Let \(M\) be an interior point of a parallelogram \(ABCD\). Prove that \[ MA + MB + MC + MD < \text{the perimeter of } ABCD . \]14The line through the circumcentre and the orthocentre of a triangle \(ABC\) (the Euler line of the triangle) crosses the interior of the side \(CA\) at a point \(M\) and the interior of the side \(CB\) …15Two circles touch each other internally at a point \(A\). Let \(AB\) be a diameter of the larger circle. Through the other endpoint \(B\) of this diameter a line is drawn which touches the smaller circle …16In the plane, two circles \(k_1\) and \(k_2\) and a line \(p\) are given. The line \(p\) cuts \(k_1\) at the points \(A\) and \(B\), and it cuts \(k_2\) at the points \(C\) and \(D\). Each of the two tangents …17Let \(k\) be a circle with centre \(O\), and let \(T\) be a point outside it. The two tangents drawn from \(T\) touch \(k\) at the points \(A\) and \(B\). Let \(k'\) be the circle with centre \(T\) that …18In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at the point \(D\). The triangle \(ADC\) is isosceles with apex \(D\), that is, \(|DA| = |DC|\). Given \(|CD| = 36\) …19A quadrilateral \(ABCD\) satisfies \[ AD = BC \qquad \text{and} \qquad \angle DAB > \angle ABC . \] Prove that then \(\angle BCD > \angle CDA\).20In a pentagon \(ABCDE\) all five sides are congruent to one another, and \[ \angle BAE = 2 \angle CAD . \] Determine \(\angle BAE\).21Let \(ABC\) be an acute triangle. The circle \(k\) with diameter \(AB\) meets the side \(AC\) at \(M\) and the side \(BC\) at \(N\). The tangents to \(k\) at \(M\) and at \(N\) meet at the point \(P\). …22Three straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …23Let \(ABC\) be a triangle and let \(a\), \(b\), \(c\) denote the lengths of the sides opposite the vertices \(A\), \(B\), \(C\) respectively. Prove that a point \(S\) is the centre of the inscribed circle …24Two equilateral triangles \(ABC\) and \(PQR\) lie in the plane so that \(R\) is an interior point of the segment \(AB\) and \(C\) is an interior point of the segment \(PQ\), the points \(A\) and \(P\) …25Let \(t_a\) and \(t_b\) be the medians of a triangle \(ABC\) drawn to the sides \(BC\) and \(CA\), and let \(P\) be the area of the triangle. Prove that \[ t_a \cdot t_b \geqslant \tfrac{3}{2} P , \] and …26Let \(ABC\) be an isosceles triangle with \(AB = BC\). A point \(M\) is chosen inside it so that \[ \angle AMC = 2 \angle ABC , \] and a point \(N\) on the segment \(AM\) satisfies \(\angle BNM = \angle ABC\). …27Circles \(k_1\) and \(k_2\) intersect at points \(P\) and \(Q\), and \(k_1\) passes through the centre of \(k_2\). Distinct points \(A\) and \(B\) lie on the arc of \(k_1\) that runs inside \(k_2\), and …28The circle inscribed in triangle \(ABC\) touches the sides \(BC\), \(CA\) and \(AB\) at the points \(D\), \(E\) and \(F\) respectively. A point \(K\) lies on the same side of the line \(EF\) as the vertex …29Convex quadrilaterals \(ABCD\) and \(PQRS\) are given, where the vertices of the quadrilateral \(PQRS\) lie on the sides or in the interior of the quadrilateral \(ABCD\). Can the sum of the diagonals of …30Find all points \(X\) inside the square \(ABCD\) for which \[ AX + CX = BX + DX . \]31Let \(ABC\) be a right triangle. Construct a point \(N\) inside \(\triangle ABC\) for which \[ \angle NBC = \angle NCA = \angle NAB. \]32On the sides of an acute triangle \(ABC\) points \(A_1 \in BC\), \(B_1 \in CA\) and \(C_1 \in AB\) are chosen so that \[ \angle CC_1B = \angle AA_1C = \angle BB_1A = \varphi, \] where \(\varphi\) is an …33In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …34Let \(H\) be the orthocentre of an acute triangle \(ABC\), and let \(A_1\), \(B_1\) and \(C_1\) be the centres of the circles circumscribed about the triangles \(BHC\), \(CHA\) and \(AHB\) respectively. …35Squares \(BCDE\), \(ACFG\) and \(BAHK\) are erected outwards on the sides of a triangle \(ABC\). After that, the parallelograms \(BKPE\) and \(CDQF\) are drawn. Prove that the triangle \(PAQ\) is right …36In triangle \(ABC\) the angles at \(A\) and \(B\) measure \(\angle A = 50^\circ\) and \(\angle B = 60^\circ\). Points \(D\) and \(E\) are taken on the sides \(AB\) and \(BC\) respectively, so that \[ \angle DCA = \angle EAC = 30^\circ . \] …37The points \(A, B, C, D, E\) lie on one circle in such a way that \(A\) and \(D\) are on opposite sides of the line \(BC\), and \(B\) and \(E\) are on opposite sides of the line \(CD\). Given that \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …38Two points \(A_1\), \(B_1\) and a line \(p\) are given in the plane. Construct a triangle \(ABC\) in which \(A_1\) is the midpoint of the side \(BC\), \(B_1\) is the midpoint of the side \(CA\), and the …39A triangle \(ABC\) has \(AB = 2\), \(BC = 3\) and \(CA = 4\). Find a polygonal line \(XYZ\) whose endpoints \(X\) and \(Z\) lie on the boundary of the triangle \(ABC\), such that \[ XY = YZ = 1 \] and …40Let \(ABC\) be a triangle with \(BC \neq CA\), and let \(H\), \(T\) and \(O\) be its orthocentre, centroid and circumcentre. Let \(P\) be the point symmetric to \(T\) with respect to \(O\), and let \(Q\) …41Let \(ABCD\) be a convex quadrilateral which is not a trapezoid. The perpendicular bisectors of the sides \(AD\) and \(BC\) meet at a point \(P\), and the perpendicular bisectors of the sides \(AB\) and …42Congruent circles \(k_1\), \(k_2\) and \(k\) all pass through a point \(P\), and each pair of them meets in one further point: \(k\) and \(k_1\) meet again at \(A\), \(k\) and \(k_2\) meet again at \(B\), …43In a triangle \(ABC\) the bisector of the angle at the vertex \(A\) meets the side \(BC\) at the point \(D\). The perpendicular dropped from \(B\) to the line \(AD\) meets the circumcircle of the triangle …44Let \(T\) be the centroid of an acute triangle \(ABC\). Let \(A'\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(A''\) be the point of the segment \(BC\) for which \[ BA' = A''C . \] …45On the sides of a triangle \(ABC\), equilateral triangles \(ADB\), \(BEC\) and \(CFA\) are constructed outwardly, so that \(D\), \(E\), \(F\) are the apexes over \(AB\), \(BC\), \(CA\) respectively. Prove …46Let \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …47A point \(D\) is chosen on the side \(BC\) of a triangle \(ABC\). Points \(E\) and \(F\), both different from \(D\), are chosen on the line \(BC\) so that \[ BE = BD \qquad \text{and} \qquad CF = CD . \] …
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