Problems

Age
Difficulty
Found: 2335

A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?

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 \(20^{\circ}\).

One of the four angles formed when two straight lines intersect is \(41^{\circ}\). What are the other three angles equal to?