# Examples for secondary school students

1. Three numbers
What is three numbers which have the property: the sum of the reciprocals of the first and second numbers is 12/7, the first and third 11/24 and the second and the third 3/8.
2. Bronze, tin and copper
Bronze is an alloy of tin and copper. An alloy of 10% tin and 90% copper is Gunmetal. If it contains 20% tin and 80% copper, it is bell metal. How many tons of molten bell metal and how many tons of copper is needed to make 100 tons of Gunmetal?
3. Points on circle
In the Cartesian coordinate system with the origin O is a sketched circle k /O; r=2 cm/. Write all the points that lie on a circle k and whose coordinates are integers. Write all the points that lie on the circle I / O; r=5 cm / and whose coordinates are
On a weekly forest brigade operates 12 students. After plant one hundred spruces get x CZK, anfter one hundred pines y CZK. How many CZK got one student for one day if planted spruces 25000 and 30000 pines week?
5. Medians of isosceles triangle
The isosceles triangle has a base ABC |AB| = 16 cm and 10 cm long arm. What are the length of medians?
6. Pyramid 8
Calculate the volume and the surface area of a regular quadrangular pyramid with the base side 9 cm and side wall with the base has an angle 75°.
7. 3s prism
It is given a regular perpendicular triangular prism with a height 19.0 cm and a base edge length 7.1 cm. Calculate the volume of the prism.
8. Trapezoid IV
In a trapezoid ABCD (AB||CD) is |AB| = 15cm |CD| = 7 cm, |AC| = 12 cm, AC is perpendicular to BC. What area has a trapezoid ABCD?
9. Truncated pyramid
How many cubic meters is volume of a regular four-side truncated pyramid with edges one meter and 60 cm and high 250 mm?
10. Eight blocks
Dana had the task to save the eight blocks of these rules: 1. Between two red cubes must be a different color. 2. Between two blue must be two different colors. 3. Between two green must be three different colors. 4. Between two yellow blocks must be fou
11. Owner of the company
The owner of the company with extensive administrative activity wants to sell the obsolete 3 machines at the price of CZK 1000 per 1 machine with a six-month warranty period. If the machine breaks down during this period, the owner will return the employ
12. Sequence - 5 members
Write first five members of the sequence ?
13. AP - basics
Determine first member and differentiate of the the following sequence: a3-a5=24 a4-2a5=61
14. Seats
Seats in the sport hall are organized so that each subsequent row has five more seats. First has 10 seats. How many seats are: a) in the eighth row b) in the eighteenth row
15. Sum 10
What is the probability that two dice throw twice in a row will result the sum of 10?
16. GP - 8 items
Determine the first eight members of a geometric progression if a9=512, q=2
17. Population
What is the population of the city with 3% annual growth, if in 10 years the city will have 60,000 residents?
18. Tetrahedral pyramid
Calculate the volume and surface of the regular tetrahedral pyramid if content area of the base is 20 cm2 and deviation angle of the side edges from the plane of the base is 60 degrees.
19. Throw
We throw 2 times with 2 dices. What is the probability that the first roll will fall more than sum of 9 and the second throw have sum 3 or does not have the sum 4?
20. Coffee
Coffee merchant has coffee robusta and arabica species. 1 kg Robusta worth 450 CZK, Arabica 1 kg is 300 CZK more expensive. Calculate how many kilograms of Robusta and Arabica will need to produce 30 kg of the mixture so that the mixture cost is 490 CZK

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