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Let the number \(\alpha\) be given by the decimal:

a) \(0.101001000100001000001 \dots\);

b) \(0.123456789101112131415 \dots\).

Will this number be rational?

Prove the irrationality of the following numbers:

a) \(\sqrt{3}{17}\)

b) \(\sqrt{2} + \sqrt{3}\)

c) \(\sqrt{2} + \sqrt{3} + \sqrt{5}\)

d) \(\sqrt{3}{3} - \sqrt{2}\)

e) \(\cos 10^{\circ}\)

f) \(\tan 10^{\circ}\)

g) \(\sin 1^{\circ}\)

h) \(\log_{2}3\)

Is it possible for

a) the sum of two rational numbers irrational?

b) the sum of two irrational numbers rational?

c) an irrational number with an irrational degree to be rational?

Prove that the number \(\sqrt {2} + \sqrt {3} + \sqrt {5} + \sqrt {7} + \sqrt {11} + \sqrt {13} + \sqrt {17}\) is irrational.

There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if

a) weights can be placed only on one side of the scales;

b) weights can be placed on both sides of the scales?

a) One person had a basement illuminated by three electric bulbs. Switches of these bulbs are located outside the basement, so that having switched on any of the switches, the owner has to go down to the basement to see which lamp switches on. One day he came up with a way to determine for each switch which bulb it switched on, descending into the basement exactly once. What is the method?

b) If he goes down to the basement exactly twice, how many bulbs can he identify the switches for?