45 candies cost the same amount of rubles as they can be bought for 20 rubles. How many sweets can you buy for 50 rubles?
Answer: 75 candies.
Solution. Let be x- the cost of one candy in rubles. Then 45 x= 20/x, where x= 2/3. Then for 50 rubles you can buy 50 / x= 75 candies.
Criteria.
Equation 45 is correct x= 20/x, but when solving it or in the future, an arithmetic error was made: 5 points.
The solution states that the price of one candy is equal to 2/3, it is checked that such a cost fits the condition of the problem, and the correct answer is obtained: 4 points.
Only the correct answer is given: 1 point.
Task 2. (7 points)
Zhenya arranged the numbers from 1 to 10 in a certain order, and Dima entered their sum in each interval between the numbers. Could it be that all the numbers written by Dima turned out to be different?
Answer: It could.
An example of the arrangement of numbers is shown below.
Criteria. Any correct decision: 7 points.
Only the correct answer is given or the correct answer and the wrong example: 0 points.
Task 3. (7 points)
Is it possible in some cells of table 8 × 8 write ones, and in the rest - zeros, so that in all columns there is a different amount, and in all lines - the same?
Answer: can.
Solution. Let the sum of the numbers in each line be x... Then the sum of all numbers in the table is 8 x, that is, the total is divided by 8. Note that the columns can contain from 0 to 8 units. The sum of all numbers from 0 to 8 is 36. To get a multiple of 8, you need to subtract 4. From 36, therefore, in our example, there should not be a column in which there are exactly 4 units.
An example is shown below (there are other examples).
Criteria. Any true example, even without any explanation: 7 points.
It is proved that if the sum in all columns is non-zero, then there is no example: 4 points.
Task 4. (7 points)
The two squares have a common vertex. Find the ratio of line segments AB and CD shown in the figure.
Answer:
Solution. Let the point O- common vertex of two squares, and their sides are equal a and b... The diagonals of the squares are and respectively. Moreover, ∠ COD= ∠COB+ ∠BOD= ∠COB+ 45 ° = ∠COB+ ∠AOC= ∠AOB... Triangles AOB and COD are similar in total angles and proportional sides at that angle.
Hence, AB: CD=
Criteria. Any correct decision: 7 points.
Correctly calculated ratio is not AB To CD, a CD To AB(respectively, the answer): 7 points.
The similarity of triangles is proved AOB and COD, but there is no further conclusion or the desired relationship was found incorrectly: 6 points.
It is proved that ∠ AOB= ∠COD but no further progress: 1 point.
Only a special case is considered (for example, when the squares have a coinciding side or when the angle between some sides of two squares is 90 °): 0 points.
Only the correct answer is given: 0 points.
Task 5. (7 points)
Numbers a, b, c and d are such that a+b= c+d ≠ 0, ac= bd... Prove that a+ c= b+ d.
Solution. If a ≠ 0, then substitute c= b d / a, we get
From here b= c and a+ c= b+ d.
If a= 0, then b ≠ 0 (otherwise a+ b= 0), therefore d= 0 (from ac= bd). But then the equality a+ b= c+ d is rewritten as b= c, whence the desired equality follows.
Other solutions are also possible.
Criteria. Any correct decision: 7 points.
The correct solution considers an expression of the form bd a(or any similar), but the case of equality of the denominator to zero is not considered: 5 points.
It is proved that ( a+c) 2 = (b+d) 2, but the case ( a+c) = — (b+ d): 3 points.
Only the case of specific numerical values is considered a, b, c, d: 0 points.
Task 6. (7 points)
There are 60 road signs along the road. Each of them contains the sum of the distances to the remaining 59 characters. Is it possible that 60 different natural numbers are written on the signs? (The distances between the characters are not necessarily whole.)
Answer: Impossible.
Solution. Let us number the signs sequentially with numbers from 1 to 60. Let us prove that the numbers written on the signs numbered 30 and 31 coincide.
Divide the remaining signs into pairs of the form k and k+ 31: 1 and 32, 2 and 33,. ... ... , 29 and 60. Note that the sum of the distances from both sign 30 and sign 31 to signs of one pair k and k+ 31 is equal to the distance between signs k and k+ 31. Since the number on the signs 30 and 31 is equal to the sum of the distances to the signs of all 29 pairs and the distance between the signs 30 and 31, the numbers on the signs 30 and 31 coincide.
Criteria. Any correct decision: 7 points.
It is stated, but not proven, that the numbers written on the two middle columns (on columns 30 and 31) are equal: 2 points.
On the example of special cases, it is shown that there will definitely be equal numbers: 0 points.
Only the correct answer is given: 0 points.
Some Pole liked my felt work, I went to look at his photographs. I saw the clean, tidy roads of Poland, wiped away a tight tear. Well, see for yourself:
October, Bialystok. There is no dirt or dirt on Polish roads. In rural areas, paths are laid for pedestrians along local roads. They are clean, flat and without holes. Mystic huh? Even in the Moscow region we have a problem with paths for people, we are not talking about other regions at all.
Here you can see the road that runs in the suburbs. I really like this mesh that prevents people from running deer across the road. These nets are vital on Moscow avenues. But for some reason I met them only in Krylatskoye (this is a special area, there are purely their own rules for organizing life) and a little in the area of Vernadsky Ave., a scanty patch.
I also like the high curb, which physically separates the different streams of cars. But in Moscow, on the contrary, they are trying to get rid of thread separators. Are we not following the plan for the cemeteries? I understand that there are 7 billion of us on Earth, that we multiply very quickly. But it is better to educate people in the field of contraception than to increase their mortality in this way.
And pay attention to the striking markings. In Poland, they save on night lighting of highways, but this is more than compensated by the excellent markings made with reflective paint. Visibility is excellent, signs are cleaned, washed, reflectors on bumpers are restored in case of loss. At night on the roads of Poland, white, red and blue colors help the driver, creating a cross between a disco and a runway.
This is what Polish roads look like at night.
This is the highway.
Perfectly smooth asphalt. Great bright markings. It is voluminous and makes noise when you cross it. Well wakes up a sleepy or absent-minded driver.
The quality of the reflective paint is clearly visible here. Again, evaluate the road surface.
Such paint is very far visible.
Russia, apparently, has its own technology for creating roadside signs. Or road services do not simply clean and change them.
Signs are clearly visible in the distance.
As a reminder, this is a local rural road. Do you remember what the Russian federal highway "Kiev" looks like, passing through the Pskov region? I hope you will not have a reason to collect potholes there. And what about the local roads of the Tver province? They prefer to drive on the side of the road, not on the asphalt.
In the case of repairs, a whole battery of striped posts is placed along the road. They are perfectly visible from afar. Unfortunately, this is not the case in Russia.
Would you like to show you how road repairs are fenced off in Russia? Gorkovskoe highway, driving towards Vladimir.
There are dirty red and gray plastic things along the track. At night they are not visible at all. For example, such plastic crap without proper lighting was simply crumpled and scattered by trucks along the Moscow-Peter highway. Imagine, it's night, there are no markings on the road, there are no signs of repair or detour, and then a kicked-off piece of crumpled plastic flies into your car.
Funny posts between the streams of cars. The one who designed them and ordered them to be installed probably naively believes that the post will be able to stop a rushing car. No, the posts are simply demolished. Drivers don't care.
Now take a look at how the problem of safe car traffic separation in Germany has been solved. The highway is being repaired. The yellow markings are temporary. The streams have a speed of 60 km / h. A high metal obstacle is installed between the streams.
And this is a Russian road of local importance. From Polenovo (Tula region) we go to the highway. The asphalt pavement leaves much to be desired, there are no markings, fog bollards and the whole roadside is covered in mud (trucks pick it up very well in the form of dust).
Tver province. Local road towards Kalyazin. Rather, the direction. There was nothing left of the road except traces of asphalt.
Even trucks go around the side of the road.
Sergei Krivoshein, head of the branch in the city of Magadan, Uprdor Kolyma, answers the question of the concerned Magadan residents about the legality of deforestation near the Kolyma federal highway on the Magadan-Sokol section.
Recently, several residents of Magadan and its environs contacted the editorial office with a request to clarify the following issue.
“In the area of the 1990th km of the Kolyma highway, between Magadan and Sokol, an outrage is happening: the forest is being cut down! - Andrey Pryakhin from Magadan asserted very emotionally. - Directly along the road, in whole glades!
Larches and poplars are then cleared of branches and transported somewhere, and the branches are dumped in the nearest quarry. "
“Who does this and by what right? - Nina Tikhonova was perplexed. - Who got in the way of the trees? As soon as the leaves have blossomed, such beauty, and this beauty - at the root! Where does the forestry protection look! Find out whose handiwork it is, and write about them! "
It didn't take long to find out: the situation was explained by the head of the branch in the city of Magadan PKU Uprdor "Kolyma" Sergei Krivoshein.
- On the section 1990-1995 km of the federal highway "Kolyma", work has begun on overhaul, - said Sergei Alekseevich. - The length of the overhaul section is 4.2 km.
At the moment, on this site, deforestation is being carried out in the right-of-way and in the roadside for the arrangement of temporary detours in those places where the roadbed will be opened for the overhaul of culverts.
Temporary road signs announcing that major repairs are being carried out on this section have been installed. Perhaps not all Magadan residents noticed them, since some hooligans got in the habit of knocking over signs at night. Every morning the road workers put them in place. In principle, if a forest is cut near the road, this is normal. This is called “clearing the right-of-way”, for example to open closed bends.
According to the standard, there should not be any vegetation within the right-of-way, and we are striving for this. Another thing is that we do not always have time. So nothing supernatural happens on our track, - assured S.A. Krivoshein.
Taking this opportunity, we asked the head of the branch about the full scope of repairs planned this summer on the federal highway passing through the territory of the Magadan Region.
- If we talk about the capital type of pavement, namely about the areas with asphalt pavement, then we have planned two of them this year. You already know one - overhaul of 1990-1995 km, the second - 2017-2021 km (near the stele at the entrance to Magadan). Renovation work will be carried out there.
In addition, we are carrying out a major overhaul of the bridge on the Razvalny brook, where in 2014, under the influence of the passage of "heavyweights", the slabs of the superstructures collapsed.
With the support of the Federal Road Agency, we were able to start work on major repairs as soon as possible: less than two years passed from the intention to carry out major repairs of the bridge to the start of these works, - noted S.A. Krivoshein. - This time was enough for us to develop design and estimate documentation, wait for a positive conclusion of the state examination, include the project in the state program, receive funding from Rosavtodor and start work from the beginning of summer.
Also this year, as part of the overhaul, we are rebuilding 18 culverts and repairing four sections with a transitional type of coverage: three sections with a length of 16 km and one with a length of 8 km.
Regarding the amount of funding for road repairs and overhauls in 2016, Sergei Krivoshein said that compared to 2015, it increased by about 10 percent and is currently just over 1 billion rubles.
Recently, the size of road signs in Russia was proposed to be reduced by as much as 30%. According to members of the government commission on traffic rules, such a decision will optimize the urban environment and save budget funds. In addition, the installation process will be simplified. At the same time, officials believe that such an innovation will not bring harm. The idea was supported by the traffic police, therefore, road signs of reduced size will begin to be introduced on Russian roads by the end of the year. After an experiment in several cities, this practice will be disseminated throughout Russia. ((material_120762)) However, Muscovites were able to try out small signs in action at the beginning of the year. Reduced road signs appeared on 5 streets of the capital in January. Instead of the standard GOST signs “Stopping prohibited” and “Parking” with dimensions of 700x700 mm, signs of 500x500 mm appeared on 3 streets, and on two even less - 400x400 mm. The "Paid Services" sign was reduced from 700x350 mm to 500x250 mm. In St. Petersburg, a pilot project on the use of international standard road signs was also launched, which will last 4 months. In agreement with the regional traffic police, 190 new road signs were placed on the street. Mayakovsky, st. Zhukovsky and st. Nekrasov from Liteiny prospect to Ligovsky prospect. The traffic organization scheme remained the same. The main goal of the experiment is to determine the feasibility of introducing new standard sizes of road signs for the development of the urban environment and subsequent savings in budgetary funds; assessment of safety performance and throughput; study of the perception of these road signs by road users in conditions of dense historical buildings. The study of the visibility of road signs was carried out back in 2014-2015, based on the results, a proposal was developed to amend the GOSTs. There is an option to reduce signs 3.27 "No Stopping" and 6.4 "Parking" from 700 x 700 mm to 500 x 500 mm, and sign 8.8 "Paid Services" - from 700 x 350 mm to 500 x 250 mm. The State Public Institution "Center for Traffic Management" argues that reduced road signs will better fit into the urban architecture and will cause less inconvenience to pedestrians, since they will be installed higher and closer to the roadway. Legislation will also be changed under reduced signs: by the end of the year, changes will be made to state standards, after which such signs can be installed throughout Russia. But whether it really will save regional budgets is a moot point It would be logical and correct to make the transition taking into account these norms - not everywhere within 1-2 years, but as the planned end of the service life of old signs. This would save money)) Also, according to the expert, in the future it would be worth changing the interval between updates of signs in GOST, increasing it to 10 years. Modern materials that can be used in manufacturing now will allow them to operate for a longer period. Moreover, there are still signs on the roads (especially in rural areas) that are 20 or more years old, while they still perform their functions. Another problem with reduced signs on the roads is their poor visibility. Situations when bushes along the roads cover the signs are not even worth disassembling: with a reduced format, they will simply not be visible. Another thing is that it is possible to place reduced road signs only in areas where the real speed of cars will not exceed 60 km / h. The fact is that the perception of road signs by drivers depends on weather conditions, as well as on the speed of movement. The higher the permitted speed, the larger the road sign should be. It is not surprising that in European cities the signs are small, because the permitted speed there, as a rule, is limited to 50 km / h. In Russia, in fact, they drive 80 km / h. ((Person_51_I am impressed by the assurances of MADI experts who conducted a study of the visibility of reduced-size road signs. They say that their readability for drivers will not deteriorate. However, in my opinion, with the introduction of a new format of signs nevertheless, it is also advisable to monitor the traffic situation, track the statistics of accidents and, based on the analysis, continue to introduce new technical means that are in demand for drivers, designed to further improve road traffic regulation)) In urban conditions, where the speeds are low, perhaps a decrease in signs makes sense. As for driving on the motorway, where the speed reaches 110 km / h, and sometimes up to 130 km / h is allowed, it is appropriate to install small signs only on test sites. Moreover, they should be regularly monitored for accidents. Also in Russia they supported the decision to reduce the total number of signs on the streets. We are talking about auxiliary signs, for example, the number of parking spaces and various additional information. True, the timing of the implementation of this project and the dismantling of the plates have not yet been named.