# The most difficult problems

1. Two hemispheres
In a wooden hemisphere with a radius r = 1, a hemispherical depression with a radius r/2 was created so that the bases of both hemispheres lie in the same plane. What is the surface of the created body (including the surface of the depression)?
2. All use computer
It is reported that 72% of working women use computers at work. Choose 3 women at random, find the probability that all 3 women use a computer in their jobs.
3. The modulus
Find the modulus of the complex number 2 + 5i
4. The half life
The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. Starting with 145 grams of a radioactive isotope, how much will be left after 3 half-lives?
5. Half of halves
Half of the square we cut off, then half of the rest, etc. Five cuts we made in this way. What part of the content of the original square is the content of the cut part?
6. Probability
How probable is a randomly selected three-digit number divisible by five or seven?
7. Brownies
Mrs. Merritt made brownies, the recipe had to be cut in 1/2. The recipe called for 2 5/8 cups of sugar. How much sugar did she use?
8. What is bigger?
Which ball has a larger volume: a football with a circumference of 66 cm or a volleyball with a diameter of 20 cm?
9. Annulus from triangle
Calculate the content of the area bounded by a circle circumscribed and a circle inscribed by a triangle with sides a = 25mm, b = 29mm, c = 36mm
10. A map
A map with a scale of 1: 5,000 shows a rectangular field with an area of 18 ha. The length of the field is three times its width. The area of the field on the map is 72 cm square. What is the actual length and width of the field?
11. The diameter
The diameter of a circle is 4 feet. What is the circle's circumference?
12. The Earth
The Earth's surface is 510,000,000 km2. Calculates the radius, equator length, and volume of the Earth, assuming the Earth has the shape of a sphere.
13. Wall and body diagonals
The block/cuboid has dimensions a = 4cm, b = 3cm and c = 12cm. Calculates the length of the wall and body diagonals.
14. Black diamond run
Taleah is skiing down a black diamond run. She begins skiing at the top of a ski trail whose elevation is about 8625 feet. The ski run ends toward the base of the mountain at 3800 feet. The horizontal distance between these two points is about 4775 feet.
15. Simplest form
Find the simplest form of the following expression: 3 to the 2nd power - 1/4 to the 2nd power
16. Compound interest 4
3600 dollars is placed in an account with an annual interest rate of 9%. How much will be in the account after 25 years, to the nearest cent?
Using one of the following forms x+p=q or px=q write an to represent these problems using x as the unknown variable Emily can jump twice as far as Evan on the broad standing board if Emily can jump 6.5 feet. How many feet can Evan jump?
18. Triangular prism - regular
The regular triangular prism is 7 cm high. Its base is an equilateral triangle whose height is 3 cm. Calculate the surface and volume of this prism.
19. A swiming
A swiming pool holds 30000lt of water. How many gallons does it hold? 1 gallon= 4.55lt
20. Compute 4
Compute the exact value of the area of the triangle with sides 14 mi, 12 mi, and 12 mi long.

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