The entrance to higher education institutions depends on this examination.

In 2018 in the Dutch state examination VWO appeared a test for an unusual topic – bitcoin. Students are told about the principle of the crypto currency and describe the process of mining, and then they are asked to solve mathematical problems about bitcoin. The first unusual section in the state exam was noticedby Reddit users.

Bitcoin is a digital currency that exists only online. It appeared on January 1, 2009 and can be used for payment in online stores or other online services. Bitcoin is not issued by the central bank, as ordinary money. It is created thanks to computers working on solving a certain mathematical problem.

It works like this: anyone can run special software on his or her computer to join the solution of this mathematical problem and get a certain reward.

Based on this information, students are asked to respond to five assignments related to the mathematical problems of crypto currency. The user Reddit with the nickname reginarhs translated the tasks into English and wrote the ways to solve them. It is not known whether this will help students.

#### What questions about bitcoin are asked in the Dutch version of the USE

- The owner of the computer, which will find a solution to the mathematical problem, receives 25 bitcoins as a reward. Since in 2014 this problem was solved every 10 minutes, 25 bitcoins appeared every 10 minutes. As of January 1 [2014], about 12, 2 million bitcoins were produced. In which year the volume of bitcoins will reach 18 million units, given that the speed with which they appear does not change.
- In reality, there are not 25 bitcoins in 10 minutes. This ratio is constantly decreasing. For 4 years from January 2009 to January 2013, the award was equal to 50 bitcoins. Since then, it has been halved every four years. Starting from what year the cost of the award for the production of bitcoins will be less than one bitcoin?
- The total volume of bitcoins that can be released is finite. In part, this is due to a reduction in the reward for mining by half every four years. The number of bitcoins currently in circulation can be calculated using the equation: C = 21 – 21 * 0.5 ^ (0.25 * t), where C is the number of bitcoins in millions, and t is the number of years since January 2009. Count the maximum amount of bitcoins that can be released.
- To regulate the total volume of released bitcoins, not only does the reward for extraction decrease, but at the same time the complexity of mathematical problems grows. This is necessary, as more and more people use their computers to solve these problems [mining]. The complexity of the problem grows exponentially according to the equation: D = 3.65 * e ^ (0.533 * t), where D is the complexity of the problem being solved, and t is the time in months, starting from January 1, 2013. Write an equation for the derivative D and prove that D is an increasing function that grows with increasing speed [of mining].
- The above formula D can be rewritten in such a way that it is possible to find the number of months remaining to a specific value of D. Rewrite the formula so that it can be done.

VWO is the final state examination in the Netherlands, based on the results of which students enter higher education institutions.