MathDB

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Part of 2005 Estonia National Olympiad

Problems(4)

altitude of hypotenuse in a right triangle, computational

Source: 2005 Estonia National Olympiad Final Round grade 9 p1

4/13/2020
The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of 9:19: 1 and two triangles of the starting triangle with a difference of areas of 4848 cm2^2. Find the original triangle sidelengths.
geometryright triangleareas
12 pieces of pizza for 7 brothers

Source: 2005 Estonia National Olympiad Final Round grade 10 p1

4/13/2020
Seven brothers bought a round pizza and cut it 1212 piece as shown in the figure. Of the six elder brothers, each took one piece of the shape of an equilateral triangle, the remaining 66 edge pieces by the older brothers did not want, was given to the youngest brother. Did the youngest brother get it more or less a seal than his every older brother? https://cdn.artofproblemsolving.com/attachments/0/7/2efaec7dab171b8bb239dc8eb282947a5c44b0.png
geometry
cos 2x = cos 2y if sin x + cos y = 1 and cos x + sin y = -1

Source: 2005 Estonia National Olympiad Final Round grade 11 p1

4/13/2020
Real numbers xx and yy satisfy the system of equalities {sin⁡x+cos⁡y=1cos⁡x+sin⁡y=−1\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases} Prove that cos⁡2x=cos⁡2y\cos 2x = \cos 2y.
trigonometrysystem of equationsalgebra
6 holes from punches in the buses of a certain bus company

Source: 2005 Estonia National Olympiad Final Round grade 12 p1

4/13/2020
Punches in the buses of a certain bus company always cut exactly six holes into the ticket. The possible locations of the holes form a 3×33 \times 3 table as shown in the figure. Mr. Freerider wants to put together a collection of tickets such that, for any combination of punch holes, he would have a ticket with the same combination in his collection. The ticket can be viewed both from the front and from the back. Find the smallest number of tickets in such a collection. https://cdn.artofproblemsolving.com/attachments/b/b/de5f09317a9a109fbecccecdc033de18217806.png
combinatorics