A straight corridor of length 100 m is covered with 20 rugs that have a total length of 1 km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?
A circle is divided up by the points \(A, B, C, D\) so that \({\smile}{AB}:{\smile}{BC}:{\smile}{CD}:{\smile}{DA} = 2: 3: 5: 6\). The chords \(AC\) and \(BD\) intersect at point \(M\). Find the angle \(AMB\).
A circle is divided up by the points \(A\), \(B\), \(C\), \(D\) so that \({\smile}{AB}:{\smile}{BC}:{\smile}{CD}:{\smile}{DA} = 3: 2: 13: 7\). The chords \(AD\) and \(BC\) are continued until their intersection at point \(M\). Find the angle \(AMB\).
The angles of a triangle are in the ratio \(2: 3: 4\). Find the ratio of the outer angles of the triangle.
One angle of a triangle is equal to the sum of its other two angles. Prove that the triangle is right-angled.
Prove that the segment connecting the vertex of an isosceles triangle to a point lying on the base is no greater than the lateral side of the triangle.
Ten straight lines are drawn through a point on a plane cutting the plane into angles.
Prove that at least one of these angles is less than \(20^{\circ}\).
In an isosceles triangle, the sides are equal to either 3 or 7. Which side length is the base?
In the acute-angled triangle \(ABC\), the heights \(AA_1\) and \(BB_1\) are drawn. Prove that \(A_1C \times BC = B_1C \times AC\).
Let \(AA_1\) and \(BB_1\) be the heights of the triangle \(ABC\). Prove that the triangles \(A_1B_1C\) and \(ABC\) are similar.