Practice library
Problems
1Two circles that do not intersect are given. Construct all of their common tangent lines.Open2The lengths of the sides of a triangle \(ABC\) are three consecutive natural numbers. The median drawn from \(A\) is perpendicular to the bisector of the angle \(\angle ABC\). Determine the lengths of …3The circles \(k_1\) and \(k_2\) meet at two points \(A\) and \(B\). Through \(A\) and through \(B\) two parallel lines are drawn. They meet the circle \(k_1\) for a second time at the points \(C\) and …4The squares \(ABCD\) and \(EFGH\) both have side \(1\). The first is cut into nine congruent small squares and the second into sixteen congruent small squares; in every small square the inscribed circle …5Every edge of a certain pyramid has length \(5\) cm, and the pyramid has five vertices in all, so its base is a square. All five vertices are sliced off, each by a single plane, and the cuts are made so …6A convex pentagon \(A_1A_2A_3A_4A_5\) is given. Let \(B_1\), \(B_2\), \(B_3\), \(B_4\) be the midpoints of the sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_5\), in that order, and let \(M\) be the …7In a triangle \(ABC\) the angle at \(A\) measures \(60^\circ\). Write \(a\), \(b\), \(c\) for the lengths of the sides \(BC\), \(CA\), \(AB\). Prove that the area of the triangle equals \[ \frac{\sqrt{3}}{4}\left(a^2 - (b-c)^2\right) . \] …8A circle is drawn through two vertices of a triangle and through the orthocentre of that triangle. Prove that this circle has the same radius as the circle circumscribed about the triangle.9Let \(H\) be the orthocenter of a triangle \(ABC\), and let \(K\) be the point symmetric to \(H\) with respect to the midpoint of the side \(BC\). Prove that \(AK\) is a diameter of the circumcircle of …10A pentagon \(ABCDE\) is inscribed in a circle. Let \(F\), \(G\), \(H\) and \(I\) be the midpoints of the segments \(BC\), \(CD\), \(DE\) and \(EA\) respectively. The lines \(FG\) and \(HI\) meet at the …11Let \(ABC\) be a triangle and let \(X\) be a point of its plane. Denote by \(A'\), \(B'\), \(C'\) the images of \(A\), \(B\), \(C\) under the reflection in the point \(X\). Let \(M\), \(N\), \(P\) be the …12Over each side of a convex quadrilateral, as a diameter, a circle is constructed. Prove that these four circles cover the quadrilateral.13Let \(M\) and \(N\) be two distinct points, neither of which lies on a given line \(p\). Construct a triangle \(ABC\) whose side \(AB\) lies on \(p\) and for which \(M\) and \(N\) are the feet of the altitudes …14The figure shows the route a hare ran while a wolf was chasing it through the fog. The hare first ran east; then it turned right, after a while it turned left, and shortly afterwards it turned left once …15Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.16The 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 …17Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.18For 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.19Two 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 …20In 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 …21The 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?22On 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 …23At 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 \(n - 1\) …
Showing 23 of 651 - problem statements are free for everyone.