Problems

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?

Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.

A circle is divided up by the points $A,B,C,D$ so that $⌣AB:⌣BC:⌣CD:⌣DA=2:3:5:6$. 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 $⌣AB:⌣BC:⌣CD:⌣DA=3:2:13:7$. 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?

In an isosceles triangle, the sides are equal to either 3 or 7. Which side length is the base?