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In memory of György Geréb, psychologist and college professor
1. A trapezium has both an inscribed circle and a circumscribed circle. One of the parallel sides is 10 units long and the radius of the incircle is units. What percentage of the area of the trapezium is the area of the incircle?
2. Two cities are apart. A car takes 1 hour less to cover this distance than another car because its speed is greater than that of the other one. Find the speeds of the two cars.
3. The points and lie on the graph of the logarithm function , (, ). Determine the values of and and the equation of the line .
4. Find the domain and range of the expression
5. Find the equation of the circle of radius passing through the point and touched from the outside by the circle of equation
6. Solve the following inequality on the set of real numbers:
7. A convex hexagon with three sides of length and three sides of length is inscribed in a circle of radius . Find the area of the hexagon.
8. The first term of a sequence is , and for . Express (in closed form) the th term of the sequence and the product of the first terms.