[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 112
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ОК, задачка-то несложная.
Число из 81 единиц 11...11 = 111111111 * (1 + 1*10^9 + 1*10^18 +...+ 1*10^72) = 111111111 * Число_1.
Число_1 = это число, состоящее из 9 единиц, густо разбавленных нулями.
Получается, что оба, 111111111 и Число_1, делятся на 9, т.е. произведение - на 81.
Well, that's a hell of a way to go... :)
I thought you were thinking hard.
Next: Solve the puzzle ONE + ONE = MANY (same letters mean the same digits, different letters mean different digits).
2000*x1+200*x2+20*x3+2*х4=10000*x5+1000*x4+100*х1+10*х6+x1
2000*x1+200*x2+20*x3+2*х4-10000*x5-1000*x4-100*х1-10*х6-x1=0
1899*х1+200*х2+20*х3-998*х4-10000*х5-10*х6=0
11394+1600+40-2994-10000-40=0
x1=6 x2=8 x3=2 x4=3 x5=1 x6=4 O=x1 E=x2 I=x3 H=x4 M=x5 G=x6
A child asked me a simple question about what boilers are made of.
The inside is ceramic. Solid metals dissolve in liquid metals. An example is a soldering iron tip (copper dissolves in solder), the tip shrinks over time and a hole is formed at the end. And not boilers, but furnaces.
Can a fly fly in a vacuum if the 'biology' of the fly is neglected.
Maybe if the fly uses the jet propulsion of the pooka. :)
The inside is ceramic. Solid metals dissolve in liquid metals. An example is a soldering iron sting (copper dissolves in solder), the sting shrinks over time and a hole is formed at its end. And not boilers, but furnaces.
I don't know if refractory bricks are ceramic or not, but they are lined inside both furnaces and boilers.
Boilers do have a place, if that is what you call converters in which steel is melted.
Are refractory bricks related to ceramics
Yes. So is Chinese "porcelain" :)
It is relatively clear there :)
Problem about an aeroplane on a transporter: An aeroplane (jet or propeller) stands on a runway with a movable covering (like a transporter). The runway can move against the plane's take-off direction, i.e. towards it. It has a control system that monitors and adjusts the speed of the runway so that the speed of the wheels of the aircraft is equal to the speed of the runway.
What to measure and relative to what is not prescribed. But! The web speed has to be "adjusted to the speed of the wheels" . So it's not referring to their relative speed - it's really zero, there's nothing to adjust. And not relative to the aircraft - we have it in the "unknowns".
The speed of rotation of the wheels is compared to linear velocity - so you have to adjust it to linear velocity. Relative to what? Relative to the axle. Not the web - we already decided above that it is 0. And not the ground (in the unknowns, together with the web/plane).
And these are non-contradictory and self-sufficient assumptions, and by that they are better, and therefore more correct, than other assumptions.
Your highlighted red phrase is completely inconsistent with the highlighted black phrase in the problem statement.
1. The problem statement says "so that", not "should" and "depending on". Be aware of the difference.
2. As we have already agreed, the speed of rotation of the wheels (if by that we mean their linear = circumferential speed) is always equal to the speed of the web. Implying anything else would render the task meaningless.
3. The conveyor belt has only one velocity - relative to the ground. The speed of the bottom point of the wheel relative to the ground is the same number.
4 There is still the speed of the aircraft. If we are talking about its speed relative to the ground, it is all clear, it is always measured that way. Nobody needs its speed relative to the ground or anything else. So it's all clear with the speed of the plane, too. The speed of the wheel axle relative to the ground is the same as the speed of the plane. But relative to the web, it can easily be calculated from the values of aeroplane speed and web speed (both relative to the ground, of course).
Thus, the problem statement is clear (if, of course, one wants to understand, rather than look for errors). If the last sentence of the condition is written as "It has a control system that monitors and adjusts the speed of the web so that it is equal to the speed of the aircraft and directed in the opposite direction", then it is clear that both speeds are relative to the ground and the condition becomes correct. And the problem in this case has the above-mentioned solution.
But even if you take, as suggested, the linear velocity of the wheel relative to the axis, it will not change anything. For the simple reason that the speed of the web (and hence the speed of the wheel), provided there is no friction and moment of inertia in the wheels, plays no role.
It was a bit surprising to see your reaction, Sergei, Vladislav, where, for example, he was taught to equate quantities with different units so easily, I don't know. Maybe you need a break, maybe to trade, to increase the deposit. I'm starting to worry about your brain. It might burst and stain your surroundings.
Yuri, I can't stand show-offs, especially on a level playing field with such a sick ego and conceit. Therefore, I won't bother you. Enjoy it.
Sergey, you have been the most hurt in this discussion. I understand that your "ego and self-esteem" has nothing to do with it, so the more interesting question is why?
And I also wonder why during the whole discussion you have never once perceived or responded to a single physical argument. On the contrary, you were only busy catching someone up on something.
And using the results of "voting" as an argument - waaaayyyyy.
And now, finally, the apotheosis - the switch to personalities.
Is it worth getting so worked up over nothing, Sergei?
Richie, I knew you were good at puzzles. These are all the solutions? There are three equations and six unknowns.
P.S. On the Mechmatov forum the discussion about the limit lim ( ln ( 2 + sqrt(arctg ( x) ⋅ sin ( 1/ x ) )), x → 0 ) is not over yet; as arguments they started mentioning Hausdorff topological spaces, which I know nothing about. But people, apart from the remaining two (me along with one other person), think it should be acknowledged that a limit does exist.
I have a big request for Farnsworth and lea. Please check, if it's not a bummer, this limit on the same packages as before (Mathematica, Maple, MathCad - on all three):
The sine argument is 1/x, and the limit itself is taken to the right of zero.