# Examples for secondary school students

#### Number of problems found: 1977

- Proof PT

Can you easy prove Pythagoras theorem using Euclidean theorems? If so, do it. - Described

Calculate perimeter of the circle described by a triangle with sides 478, 255, 352. - Newton's task

Grass grows in the meadow equally fast and evenly. It is known that 99 cows graze meadow for 14 days and 95 cows by 22 days. How many cows graze meadow for 77 days? - Area of RT

In the right triangle has orthogonal projections of legs to the hypotenuse lengths 7 cm and 12 cm. Determine the area of this triangle. - Wire D

Wire length 1 m is bent so that it forms a semicircle circuit (including the diameter). Determine the radius of the semicircle. - Square

Points A[-9,7] and B[-4,-5] are adjacent vertices of the square ABCD. Calculate the area of the square ABCD. - Square 2

Points D[10,-8] and B[4,5] are opposed vertices of the square ABCD. Calculate area of the square ABCD. - Triangle TBC

TBC is isosceles triangle with base TB with base angle 63° and legs length |TC| = |BC| = 25. How long is the base TB? - Gimli Glider

Aircraft Boeing 767 lose both engines at 42000 feet. The plane captain maintain optimum gliding conditions. Every minute, lose 1910 feet and maintain constant speed 211 knots. Calculate how long it takes to plane from engine failure to hit the ground. Cal - Cubes

One cube is inscribed sphere and the other one described. Calculate difference of volumes of cubes, if the difference of surfaces in 257 mm^{2}. - Forces

In point O acts three orthogonal forces: F_{1}= 20 N, F_{2}= 7 N and F_{3}= 19 N. Determine the resultant of F and the angles between F and forces F_{1}, F_{2}and F_{3}. - Axial section

Axial section of the cone is an equilateral triangle with area 168 cm^{2}. Calculate the volume of the cone. - Diagonal

Determine the dimensions of the cuboid, if diagonal long 53 dm has angle with one edge 42° and with other edge 64°. - Cone A2V

Surface of cone in the plane is a circular arc with central angle of 126° and area 415 cm^{2}. Calculate the volume of a cone. - Rotation

The right triangle with legs 11 cm and 18 cm rotate around the longer leg. Calculate the volume and surface area of the formed cone. - Euclid2

In right triangle ABC with right angle at C is given side a=27 and height v=12. Calculate the perimeter of the triangle. - Right

Determine angles of the right triangle with the hypotenuse c and legs a, b, if: ? - Euclid3

Calculate height and sides of the right triangle, if one leg is a = 81 cm and section of hypotenuse adjacent to the second leg c_{b}= 39 cm. - Euclid1

Right triangle has hypotenuse c = 27 cm. How large sections cuts height h_{c}=3 cm on the hypotenuse c? - Hole

In the center of the cube with edge 14 cm we will drill cylinder shape hole. Volume of the hole must be 27% of the cube. What drill diameter should be chosen?

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