# Examples for secondary school students - page 34

1. Miraculous tree Miraculous tree grows so fast that the first day increases its height by half the total height of the second day by the third, the third day by a quarter, etc. How many times will increase its height after 6 days?
2. Sum of series Determine the 6-th member and the sum of a geometric series: 5-4/1+16/5-64/25+256/125-1024/625+....
3. Pyramid in cube In a cube with edge 12 dm long we have inscribed pyramid with the apex at the center of the upper wall of the cube. Calculate the volume and surface area of the pyramid.
4. Height of the arc - formula Calculate the height of the arc if the length of the arc is 77 and chord length 40. Does exist a formula to solve this?
5. RT and circles Solve right triangle if the radius of inscribed circle is r=9 and radius of circumscribed circle is R=23.
6. Rectangle Anton Difference between length and width of the rectangle is 8. Length is 3-times larger than the width. Calculate the dimensions of the rectangle.
7. Lion or virgin We toss coin, every throw fits lion or a virgin with equal probability 1/2. Determine how much at least we have to make throws that with probability 0.9 lion fell at least once.
8. Cross-sections of a cone Cone with base radius 16 cm and height 11 cm divide by parallel planes to base into three bodies. The planes divide the height of the cone into three equal parts. Determine the volume ratio of the maximum and minimum of the resulting body.
9. Flywheel Flywheel turns 450 rev/min (RPM). Determine the magnitude of the normal acceleration of the flywheel point which are at a distance of 10 cm from the rotation axis.
10. Cuboid - volume and areas The cuboid has a volume of 250 cm3, a surface of 250 cm2 and one side 5 cm long. How do I calculate the remaining sides?
11. Candies In the box are 12 candies that look the same. Three of them are filled with nougat, five by nuts, four by cream. At least how many candies must Ivan choose to satisfy itself that the selection of two with the same filling? ?
12. Silo Outer perimeter of silo is 32 m. Concrete wall is 35 cm thick. What is diameter and area of inside floor of silo?
13. Swimming pool The pool shape of cuboid is 299 m3 full of water. Determine the dimensions of its bottom if water depth is 282 cm and one bottom dimension is 4.7 m greater than the second.
14. Medians in triangle Median of isosceles triangle has a length 3 cm. Determine the length of its sides if its perimeter is 16 cm.
15. Cube corners From cube of edge 14 cm cut off all vertices so that each cutting plane intersects the edges 1 cm from the nearest vertice. How many edges will have this body?
16. Sails We known heights 220, 165 and 132 of sail. It has triangular shape. What is the surface of the sail?
17. Right angled triangle 2 LMN is a right angled triangle with vertices at L(1,3), M(3,5) and N(6,n). Given angle LMN is 90° find n
18. Triangle ABC Triangle ABC has side lengths m-1, m-2, m-3. What has to be m to be triangle a) rectangular b) acute-angled?
19. Chess How many different ways can initiate a game of chess (first pass)?
20. The city 2 Today lives 167000 citizens in city. How many citizens can we expect in 11 years if their annual increase is 1%?

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