Six-phase star and double zigzag. School stage of the All-Russian Mathematics Olympiad for schoolchildren - document The figure shows a zigzag of 6

ABAKA-6

Numbers

Geometry

Combinatorics

Find the smallest even four-digit number that is a multiple of 7 and 23.

Divide the figure shown in the figure into four equal parts so that the cutting line goes along the sides of the squares. Come up with two solutions.

6 teams participate in the school tournament. Teams will play each other once. How many games will be played in the tournament?

How much is the sum of all the even numbers of the first two hundred greater than the sum of all the odd numbers of the first two hundred?

What is the greatest number of sides that a figure that is a common part of a triangle and a convex quadrilateral can have? Draw an example.

The box contains 40 balls of different colors: 17 green, 12 blue, 5 red; the remaining 6 balls are colored white and black. What is the smallest number of balls that must be taken out without looking into the box so that among the taken out balls there will be at least 2 blue balls?

Find the smallest natural number, when divided by , and 3.6 will be natural numbers.

Cut the shapes shown in the figure into two equal parts along the grid lines so that there is a circle in each part.

Andrey, Vasya and Serezha exchanged stamps: Vasya gave Serezha 5 of his stamps, Serezha gave Andrey 4 stamps, and Andrey gave Vasya 2 stamps. As a result, all three boys now have the same amount of marks. How many stamps did Andrei have at first, if only three friends have 30 stamps?

If from the numerator of the fraction
subtract a certain number and add it to the denominator, then after reduction you get a fraction. What number was subtracted from the numerator and added to the denominator?

We plant nine trees in the garden,

Three trees in each row.

There should be ten rows in total.

How to do it?

How many different fractions can be formed from numbers? 3, 5, 7, 11, 13, 17 So , so that each fraction includes 2 different numbers . How many of them will be

proper fractions ?

Nikita chose two three-digit numbers whose sums of digits coincide. From the larger number he took away the smaller number. What is the highest number Nikita could get?

Draw 8 points and connect them with segments so that the segments do not intersect and exactly 4 segments pass through each point

The soldiers lined up in a column one at a time, and among the soldiers there were only brunettes, blondes and redheads. There are exactly 17 brunettes with blondes breathing down their necks, and there are also exactly 23 blondes with brunettes right behind them. What is the smallest number of redheads that could be in the ranks?

Divide the number 310 into three parts x, y And z so that x : y= 3:2, and y : z=5:3. Name the value z .

The figure shows a zigzag of 6 squares 1 cm  1 cm. Its perimeter is 14 cm. What is the perimeter of a similar zigzag consisting of 2007 squares? ( Crux Mathematicorum , volume 33,N 5 )

Guests came to Anna and Bertha for the holiday. It turned out that Anna knows 80% of the guests, and Bertha - 60%. Each guest knows at least one of these girls, and 6 people are known to both of them. How many guests were at the party?

ABAKA-6

Word problems

Logics

There were two sets of freight cars at the station (all cars were the same length). The first train had 12 more cars than the second; when 4 cars were uncoupled from each train, the length of the first train turned out to be 2 times longer than the length of the second train. How many carriages were there in each train?

Three peaches, 2 pears and an apple together weigh 650 g, and 2 peaches, 3 pears and 4 apples together weigh 850 g. How much do a peach, pear and apple weigh together? Give your answer in grams.

A dog is chasing a rabbit 150 feet away. She jumps 9 feet every time the rabbit jumps 7 feet. How many jumps must a dog make to catch a rabbit?

A tailor has a piece of cloth 16 meters long, from which he cuts 2 meters every day. After how many days will he cut the last piece?

Three boys each have a certain number of apples. The first boy gives the others as many apples as each of them has. Then the second one gives the other two as many apples as each of them has; in turn, the third gives each of the two as many apples as each has at that moment. After this, it turns out that each boy has 8 apples. How many apples did each boy have at the beginning?

The starlings settled in the trees. When they sat down one by one on a tree, one starling did not have enough wood, and when two starlings sat on each tree, one tree was left unoccupied. How many starlings were there and how many trees were there?

Two people peeled 400 potatoes; one cleaned 3 pieces per minute, the other -- 2. The second one worked 25 minutes more than the first one. How long did each person work?

Two fishermen were asked: “How many fish are in your baskets?” “My basket contains half the number of fish that are in his basket, plus ten more,” answered the first. “And I have as many fish in my basket as he has, and twenty more,” said the second. How many fish does each fisherman have?

When dried, grass loses 80% of its mass. How many tons of hay will be produced from 4 tons of fresh grass? How many tons of grass need to be cut to dry 4 tons of hay?

IN Ancient Rome Legal philosophers loved to ask each other this problem. The widow is obliged to share the inheritance of 3,500 gold pieces left after her husband with the child who is about to be born. If it is a son, then the mother, according to Roman law, receives half of the son's share. If a daughter is born, the mother receives double the daughter's share. But it so happened that twins were born - a son and a daughter. How should the inheritance be divided so that all legal requirements are met?

Someone went on foot from Moscow to Kyiv. How many days will it take to reach Kyiv if it passes along miles every hour and will be on the road every day
h? From Moscow to Kyiv 855 versts.

Two mathematicians meet, the first speaks to the second.
- I know that you have three children, how old are they?
The second one begins to reason:
- Well, the product of their years is equal to 36 and the sum is equal to the number of the departing tram (They see this number).
The first thought and said:
- I don't have enough information.
2nd says:
- Well, there is an older redhead.
The first one thought and said how old they were. What did this mathematician say?

ABAKA-6. ANSWERS

Numbers

Geometry

MATHEMATICS

2009-2010 academic year

5th grade

TASKS

1. Decipher the recording:

+ to s and

Identical letters must have the same numbers, and different letters must have different numbers.

2. Find the meaning of the expression:

5. From a barrel containing at least 10 liters. gasoline, pour exactly 6 liters using a 5 liter can. and a nine-liter bucket.

School stage All-Russian Olympiad schoolchildren

MATHEMATICS

2009-2010 academic year

5th grade

Tasks with solutions and answers

1. Decipher the recording: k and s

+ to s and

Identical letters must have the same numbers, and different letters must have different numbers.

Answer: 495 + 459 = 954

2. Find the meaning of the expression:

2009 ∙ 25 – 25 ∙ 2007 + 2006 ∙ 15 – 15 ∙ 2004 + 2003 ∙ 5 – 5 ∙ 2001

Solution:

Answer: 90

3. Mom bought apples for her children - Vanya, Nina and Misha. The children had to divide the apples equally among themselves. Vanya came first, counted the apples, took the third part and left. Then Nina came and, believing that she had arrived first, counted the remaining apples, took a third of these apples, and left. Finally, Misha came and took a third of the remaining apples. After this there are 8 apples left. How many apples did the mother buy for her children?

Solution:

Answer: 27 apples

4. The figure shown in the figure consists of 7 identical squares. Its perimeter is 32 cm. Find the area of the figure.

Solution:

Since the perimeter is 32 cm, the length of one side of the square will be equal to 32:16 = 2 (cm), which means the area of seven squares is

Answer: 28 cm 2

5. From a barrel containing at least 10 liters. gasoline, pour exactly 6 liters using a 5 liter can. and a nine-liter bucket.

Solution:

Can (5 l.)
Bucket (9 l.)

Despite the wide distribution of three-phase current, in a number of important areas of technology it is impossible to do without. This applies, for example, to electrolysis plants in metallurgical plants and electric traction.

Relatively recently, direct current was obtained from motor generators. In a modern way receiving direct current is direct straightening using semiconductor (selenium, germanium, silicon) rectifiers.

The transition from motor-generators to direct rectification, in addition to design differences (rotating machines are replaced by stationary devices), has an important feature. It consists in the fact that the alternating and direct current circuits of the motor-generator are electrically isolated; in direct rectification they are connected, since the valves forming the rectifier and the secondary ones are directly connected. It is absolutely clear that the rectified current cannot be sinusoidal; it contains not only a variable, but also a constant component (see article " The concept of magnetic balance of a transformer"), which with some rectification schemes has a very adverse effect on the operation of the transformer.

With this difficult question Readers can read the book by I. L. Kaganov “Industrial Electronics”, 1968. Here, in the most general terms, the main provisions necessary to explain the types of connections of transformers feeding rectifiers are noted.

Rectifiers that convert three-phase current into direct current are three-phase current rectifiers, regardless of the number of phases of the secondary winding. Primary winding The transformer feeding the three-phase current rectifier is connected in a star, delta or zigzag and receives power from the three-phase current network. Secondary winding can be three-phase, six-phase, twelve-phase, which determines the rectification system; three-phase, six-phase, twelve-phase and so on.

Picture 1. General properties systems and rectification circuits. Rectified voltage curves ( A); rectification of single-phase current using a bridge circuit ( bd); rectification of three-phase current using a bridge circuit ( G) and according to the circuit with a zero output ( V).

In Figure 1, A from top to bottom the rectified voltage curves are shown for three-phase ( U 3), six-phase ( U 6) and twelve-phase ( U 12) straightening. This figure shows only the nature of the phenomena (and not quantitative relationships), illustrating the following:
a) the smallest ripples (waviness) are obtained with twelve-phase rectification, which is good;
b) the duration of the anode current of each phase is the highest with three-phase rectification ( t 3 > t 6 > t 12); from this position, three-phase rectification is better;
c) the average values of the rectified voltage for different rectification systems are not the same ( U 12 > U 6 > U 3).

Rectification circuits

Transformer connection diagram and forced magnetization flux

When the primary winding is connected into a star, and the secondary winding into a six-phase star, the forced magnetization flux changes direction every sixth part. It passes up through all the rods (and down through the air, since unidirectional flows cannot be closed in the yoke), and after 1/6 of the period it changes direction, passing down through all the rods and up through the air. The forced magnetization flux is triple the frequency of the supply network and is called single-phase forced magnetization flux.

Six-phase rectification when connecting the secondary windings of the transformer in a double zigzag

based on the fact that when connected in a zigzag, a forced magnetization flux does not occur 2. Each transformer rod contains: a primary winding A(B, C) and three sections of secondary windings x, a, d (y, b, e; z, c, f), which belong to different phases. Windings x, y, z form an internal star, the neutral of which is the negative pole of the rectifier. Windings are connected to the free ends of the inner star a, b, c, d, e, f, the outer ends of which feed the valves 1 – 6 . The common point to which the valves are connected serves as the positive pole of the rectifier.

Figure 2. Six-phase rectification in a star-double zigzag circuit.

The connections in Figure 2, A corresponds (Figure 2, b) (emf) of the secondary windings, from which it is clear: the sequence of operation of the valves 1 , 2 , …, 6 , value e. d.s. secondary winding E 2 (geometric difference in emf of sections of different phases), operating time of each valve is 60°.

widespread due to the following positive properties:
1. Rectified voltage ripple corresponds to six-phase rectification. This is significantly better than three-phase rectification, since the waviness is less, Figure 1, A.
2. The duration of operation of each valve is 1/3 of period 3. This is significantly better than six-phase rectification (where the valve operates for 1/6 of the period), since the transformer secondary windings and valves are more fully used.
3. The current passing through each secondary winding of the transformer and through each valve is half that in the star-double zigzag circuit (Figure 2), since in the circuit with an equalizing reactor (Figure 3) two valves operate in parallel, and in the above-mentioned circuit valves operate one at a time.
4. There is no single-phase forced magnetization current due to the fact that the rectified current passes through two phases of the secondary windings included in different groups 4.

The connection diagram of the transformer and valves is shown in Figure 3, A. Primary windings not shown in Figure 3 A, (triangle). Six secondary windings - two on each rod - form two groups. One of them is in neutral 0 1 the ends are connected, and the beginnings a, b And c output for connecting valves 1 , 3 And 5 . The other one is in neutral 0 2 beginnings are connected, and to the ends x, y And z valves are connected 2 , 4 And 6 . Between zero points 0 1 and 0 2 stars included surge reactor UR, the midpoint of which is the negative pole of the rectifier. The equalization reactor sections are connected back to back and placed on a two-rod magnetic core. The positive pole of the rectifier is the common point to which the valves are connected 1 , 3 and 5 (odd group), 2 , 4 And 6 (even group).

Figure 3. Six-phase rectification in a circuit with an equalizing reactor.

In Figure 3, b solid lines show phase voltages u 2a, u 2b and u 2c odd star secondary windings; dashed lines represent phase voltages u 2x u 2y and u 2z even star. The rectified voltage curve is highlighted with a thick line. It consists of sections (tops) of sinusoids with a six-fold repetition period and corresponds to the operating mode, which is illustrated in Figure 3. G. It shows that at the beginning of the process, the valves operate in parallel 5 And 6 . Then over time t 1 valves operate in parallel 6 And 1 . During t 2 valve 1 continues to work, but instead of a valve 6 the valve turns on 2 . Then the valves work 2 And 3 (time t 3), and then 3 And 4 (t 4) and finally 4 and 5 ( t 5). After that, everything is repeated in the same order. In short, at any moment two valves are operating in parallel, and for this it is necessary not only to connect them appropriately, but also to equalize the instantaneous voltage values in the circuits of parallel operating valves. This is exactly what the surge reactor is used for. Let's look at his work.

Suppose the valves operate in parallel 6 And 1 . Difference of instantaneous values of phase voltages u 2y and u 2a is defined by vertical lines (ordinates) u K (Figure 3, b) and is induced in two series-connected sections of the surge reactor UR.

The equalizer reactor sections are identical. Therefore the voltage u k1 = u k2 = ½ u k are equal in magnitude, but with respect to its average output they have different signs. Hence, u k2 (in our example) is added to the phase voltage u 2a (valve 1 ), but is subtracted from the phase voltage u 2y (valve 6 ). As a result, the voltages are equalized (hence the name - equalizing reactor), which ensures parallel operation of the two valves. The nature of the voltage change in the surge reactor is shown in Figure 3. V. Comparing it with Figure 3, b, it is easy to see that the frequency in the equalizing reactor is 3 times greater than the frequency of the supply network (compare the duration of the periods T/3 and T).

For the balancing reactor to operate, its magnetic core must be magnetized, and for this it is sufficient that the current passing through one of its sections reaches approximately 1% of the load current of one of the secondary circuits 5 . If the load is less than 1%, then the surge reactor does not work. In this case, instead of the double three-phase mode 6, the rectifier operates like a regular six-phase one, and the voltage instead of U’ 2max increases by 13 – 15%, reaching the value U 2max. Such an increase in voltage is not always permissible, 7 therefore either a ballast load of approximately 1% is created (but this is unprofitable at significant powers), or the equalizing reactor is artificially magnetized with a triple-frequency current. For this purpose, a secondary winding is connected to the equalizing reactor, the principle of operation of which is discussed in the article " Open triangle. Open triangle ".

1 In Fig. 1, b it is clearly visible that during one half cycle the current flows in the direction of the green arrows through the valves 1 And 2 . During the other half-cycle, the direction of the current through the valves 3 And 4 indicated by red arrows. In the three-phase circuit in Fig. 1, G in the positive half-cycle phase A current passes through valves 6, 7 and 9 (red arrows). In the positive half-cycle phase IN The direction of the current is shown by green arrows. Note the opposite direction of the green and red arrows on the secondary windings of the transformer.
2 The current passes simultaneously through two sections of secondary windings located on different rods, which corresponds to the symmetrical passage of current through two primary windings located on the same rods. Thanks to this m.d.s. on each of the rods are balanced.
3 The values given here (1/3, 1/6 of the period, and so on) correspond to an idealized picture.
4 One group of windings is connected in a star at the beginnings, and the other at the ends. This means that the currents in the secondary windings of one phase have opposite directions.
5 The load current always passes through the surge reactor, which is clearly visible from Fig. 3, A.
6 The mode is called double three-phase, since two three-phase groups of windings operate, with each secondary winding operating for 1/3 of a period, that is, the same as with three-phase rectification.
7 At night, the tram and trolleybus networks are almost not loaded, but the car lighting is turned on, and for incandescent lamps, increasing the voltage sharply reduces their service life.