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<image>The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles. [center] [/center]
|
50
| 0
|
|
<image>Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle?
|
7
| 1
|
|
<image>Consider a $1$ by $2$ by $3$ rectangular prism. Find the length of the shortest path between opposite corners $A$ and $B$ that does not leave the surface of the prism.
|
3\sqrt{2}
| 2
|
|
<image>Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. Nair build triangular figures by following these rules: $\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on. $\bullet$ The central column must be made up exclusively of small parts. $\bullet$ Outside the central column, only large pieces can be placed. Following the pattern, how many pieces will Nair use to build Figure $20$ ?
|
460
| 3
|
|
<image>The following diagram shows a grid of $36$ cells. Find the number of rectangles pictured in the diagram that contain at least three cells of the grid.
|
345
| 4
|
|
<image>In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$ , as shown. The area of $\triangle{ABG}$ is $36$ , the area of trapezoid $CFED$ is $144$ , and $AB = CD$ . Find the area of trapezoid $BGFC$ . [center][/center]
|
45
| 5
|
|
<image>Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that: $\bullet$ The area of square $KLMN$ is $1$ . $\bullet$ The vertices of the triangle divide three sides of the square up into these ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of the triangle $XYZ$ ? (Note that the sketch is not drawn to scale).
|
\frac{11}{50}
| 6
|
|
<image>A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is $12$ m. What is the area (in $m^2$ ) of the room?
|
18\pi
| 7
|
|
<image>The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles. [center][/center]
|
65
| 8
|
|
<image>In the diagram, $\angle AOB = \angle BOC$ and $\angle COD = \angle DOE = \angle EOF$ . Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$ . Find $\angle AOF$ .
|
118^\circ
| 9
|
|
<image>A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$ . The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$ . Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$ .
|
1
| 10
|
|
<image>In the rectangular parallelepiped in the figure, the lengths of the segments $EH$ , $HG$ , and $EG$ are consecutive integers. The height of the parallelepiped is $12$ . Find the volume of the parallelepiped.
|
144
| 11
|
|
<image>A surveillance service is to be installed in a park in the form of a network of stations. The stations must be connected by telephone lines so that any station can communicate with all the others, either through a direct connection or through at most one other station. Each station can be directly connected by a cable to a maximum of 3 other stations. The diagram shows an example of such a network connecting 7 stations. What is the maximum number of stations that can be connected in this way?
|
10
| 12
|
|
<image>Problem 6.7. Anya places pebbles on the sand. First, she placed one stone, then added pebbles to form a pentagon, then made a larger outer pentagon with pebbles, then another outer pentagon, and so on, as shown in the picture. The number of stones she had arranged on the first four pictures: 1, 5, 12, and 22. If she continues to create such pictures, how many stones will be on the 10th picture? -
|
145
| 13
|
|
<image>Below are five distinct points on the same line. How many rays originate from one of these five points and do not contain point $B$?
|
4
| 14
|
|
<image>Problem 8.7. Given an isosceles triangle $A B C$, where $A B=A C$ and $\angle A B C=53^{\circ}$. Point $K$ is such that $C$ is the midpoint of segment $A K$. Point $M$ is chosen such that: - $B$ and $M$ are on the same side of line $A C$; - $K M=A B$ - angle $M A K$ is the maximum possible. How many degrees does angle $B A M$ measure?
|
44
| 15
|
|
<image>In the diagram, points $A, B, C$, and $D$ are on a circle. Philippe uses a ruler to connect each pair of these points with a line segment. How many line segments does he draw?
|
6
| 16
|
|
<image>A circle is inscribed in a sector that is one sixth of a circle of radius 6 . (That is, the circle is tangent to both segments and the arc forming the sector.) Find, with proof, the radius of the small circle.
|
2
| 17
|
|
<image>Problem 6.8. There are exactly 120 ways to color five cells in a $5 \times 5$ table so that each column and each row contains exactly one colored cell. There are exactly 96 ways to color five cells in a $5 \times 5$ table without a corner cell so that each column and each row contains exactly one colored cell. How many ways are there to color five cells in a $5 \times 5$ table without two corner cells so that each column and each row contains exactly one colored cell?
|
78
| 18
|
|
<image>A surveillance service is to be installed in a park in the form of a network of stations. The stations must be connected by telephone lines so that any of the stations can communicate with all the others, either by a direct connection or through at most one other station. Each station can be directly connected by a cable to at most 3 other stations. The diagram shows an example of such a network connecting 7 stations. What is the maximum number of stations that can be connected in this way?
|
10
| 19
|
|
<image>Problem 9.6. In triangle $A B C$, the angles $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$ are known. On side $A C$, point $K$ is marked, and on side $B C$, points $L$ and $M$ are marked such that $K L=K M$ (point $L$ lies on segment $B M$). Find the length of segment $L M$, if it is known that $A K=4, B L=31, M C=3$.
|
14
| 20
|
|
<image>A circular wheel has 7 equal sections, and each of them will be painted with one of two colors. Two colorings are considered equivalent if one can be rotated to produce the other. In how many non-equivalent ways can the wheel be painted?
|
20
| 21
|
|
<image>8. Famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with diameter $P Q$. Point $M$ is the midpoint of the skateboard, $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take if it is known that the angular measure of the arc $A B$ is $24^{\circ} ?$
|
12
| 22
|
|
<image>Problem 10.3. On the side $AD$ of rectangle $ABCD$, a point $E$ is marked. On the segment $EC$, there is a point $M$ such that $AB = BM, AE = EM$. Find the length of side $BC$, given that $ED = 16, CD = 12$.
|
20
| 23
|
|
<image>7.1. Given a rectangular parallelepiped $2 \times 3 \times 2 \sqrt{3}$. What is the smallest value that the sum of the distances from an arbitrary point in space to all eight of its vertices can take?
|
20
| 24
|
|
<image>In the drawing below, \( C \) is the intersection point of \( AE \) and \( BF \). \( AB = BC \) and \( CE = CF \). If \(\angle CEF = 50^\circ\), determine the angle \(\angle ABC\).
|
20^\circ
| 25
|
|
<image>Problem 4. A square area was paved with square tiles (all tiles are the same). A total of 20 tiles adjoin the four sides of the area. How many tiles were used in total?
|
36
| 26
|
|
<image>4. In a certain city, the fare scheme for traveling by metro with a card is as follows: the first trip costs 50 rubles, and each subsequent trip costs either the same as the previous one or one ruble less. Petya spent 345 rubles on several trips, and then on several subsequent trips - another 365 rubles. How many trips did he make?
|
15
| 27
|
|
<image>1.1. Arina wrote down a number and the number of the month of her birthday, multiplied them and got 248. In which month was Arina born? Write the number of the month in your answer.
|
8
| 28
|
|
<image>B4. Calculate exactly, without using a pocket calculator: $\frac{(-2)^{-3}}{(-0.2)^{3}}-\left(\frac{2}{5}\right)^{-3} \cdot(-3)^{-2} \cdot 0.1^{-1}$. ## PROBLEMS FOR THE THIRD YEAR Before you are two sets of problems. Problems 1 to 6 of the first set are solved by selecting the correct answer from the five proposed answers and writing it in the table below the corresponding number. Only one answer is correct. A correct answer will be scored with two points, while a wrong answer will result in the deduction of one point. Problems 1 to 4 of the second set are solved on the attached paper. The solution to each of these problems will be evaluated on a scale from 0 to 6 points. Write only your code on the sheets where you will solve the problems. Write your solution clearly and legibly with a pen. The time allowed for solving is 90 minutes. THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS. ## PART I
|
-\frac{125}{72}
| 29
|
|
<image>Problem 11.4. On a horizontal floor, there are three volleyball balls with a radius of 18, each touching the other two. Above them, a tennis ball with a radius of 6 is placed, touching all three volleyball balls. Find the distance from the top point of the tennis ball to the floor. (All balls are spherical.)
|
36
| 30
|
|
<image>Problem 9.5. On the base $AC$ of isosceles triangle $ABC (AB = BC)$, a point $M$ is marked. It is known that $AM = 7, MB = 3, \angle BMC = 60^\circ$. Find the length of segment $AC$.
|
17
| 31
|
|
<image>In a trapezoid, the smaller base is equal to 2, and the adjacent angles are each $135^{\circ}$. The angle between the diagonals, facing the base, is $150^{\circ}$. Find the area of the trapezoid.
|
2
| 32
|
|
<image>Problem 8.6. For quadrilateral $ABCD$, it is known that $AB=BD, \angle ABD=\angle DBC, \angle BCD=90^{\circ}$. A point $E$ is marked on segment $BC$ such that $AD=DE$. What is the length of segment $BD$, if it is known that $BE=7, EC=5$?
|
17
| 33
|
|
<image>2.1. The numbers 7, 8, 9, 10, 11 are arranged in a row in some order. It turned out that the sum of the first three of them is 26, and the sum of the last three is 30. Determine the number standing in the middle.
|
11
| 34
|
|
<image>An isosceles triangle \( \triangle ABC \) is given with \( AB = BC \). Point \( E \) is marked on the ray \( BA \) beyond point \( A \), and point \( D \) is marked on the side \( BC \). It is known that: \[ \angle ADC = \angle AEC = 60^\circ, \quad AD = CE = 13 \] Find the length of segment \( AE \), given that \( DC = 9 \).
|
4
| 35
|
|
<image>## Task 2 - 270522 A tourist who lives in Magdeburg (M) wants to visit each of the cities Schwerin (S), Neubrandenburg (N), and Berlin (B) exactly once on a round trip and then return to his place of residence. One possible route would be from Magdeburg via Berlin, Schwerin, and Neubrandenburg back to Magdeburg (see illustration). List all the travel routes the tourist can choose under the given conditions! How many travel routes are there in total? A justification is not required.
|
6
| 36
|
|
<image>A solid triangular prism is made up of 27 identical smaller solid triangular prisms, as shown. The length of every edge of each of the smaller prisms is 1 . If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is painted?
|
\frac{1}{3}
| 37
|
|
<image>Problem 3. Senya cannot write some letters and always makes mistakes in them. In the word TETRAHEDRON he would make five mistakes, in the word DODECAHEDRON - six, and in the word ICOSAHEDRON - seven. How many mistakes would he make in the word OCTAHEDRON?
|
5
| 38
|
|
<image>The gure below shows a large square divided into $9$ congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area $14$ . Find the area of the large square.
|
63
| 39
|
|
<image>A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$ .
|
2
| 40
|
|
<image>Problem 7.7. In three of the six circles of the diagram, the numbers 4, 14, and 6 are recorded. In how many ways can natural numbers be placed in the remaining three circles so that the products of the triples of numbers along each of the three sides of the triangular diagram are the same?
|
6
| 41
|
|
<image>Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter?
|
\frac{1}{\sqrt{3}}
| 42
|
|
<image>B3. Šivilja had a piece of fabric in the shape of a square with a side length of $8 \mathrm{~cm}$. From it, she cut out two isosceles trapezoids and obtained a loop (see the image, the loop is represented by the shaded area). The lengths of the bases of the cut trapezoids are $8 \mathrm{~cm}$ and $2 \mathrm{~cm}$, and the height is $3 \mathrm{~cm}$. A What is the area of one of the cut trapezoids in square centimeters? B What is the area of the loop in square centimeters? C What fraction of the area of the square does the area of the loop represent? Express the result as a decimal number to 2 decimal places.
|
0.53
| 43
|
|
<image>Ezekiel has a rectangular piece of paper with an area of 40 . The width of the paper is more than twice the height. He folds the bottom left and top right corners at $45^{\circ}$ and creates a parallelogram with an area of 24 . What is the perimeter of the original rectangle?
|
28
| 44
|
|
<image>A game starts with seven coins aligned on a table, all with the crown face up. To win the game, you need to flip some coins in such a way that, in the end, two adjacent coins always have different faces up. The rule of the game is to flip two adjacent coins in each move. What is the minimum number of moves required to win the game?
|
4
| 45
|
|
<image>65. How many terms of the sum $$ 1+2+3+\ldots $$ are needed to obtain a three-digit number, all of whose digits are the same? 20 ## Search for solutions
|
36
| 46
|
|
<image>**p1.** $17.5\%$ of what number is $4.5\%$ of $28000$ ?**p2.** Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$ . The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p3.** In the $xy$ -plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$ . Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at?**p4.** What are the last two digits of the sum of the first $2021$ positive integers?**p5.** A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$ th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p6.** How many terms are in the arithmetic sequence $3$ , $11$ , $...$ , $779$ ?**p7.** Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$ ?**p8.** What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$ ?**p9.** Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$ , $AB = 10$ , $BC = 9$ , and the area of $\vartriangle ABC$ is $36$ . Compute the length of $AC$ . **p10.** If $x + y - xy = 4$ , and $x$ and $y$ are integers, compute the sum of all possible values of $ x + y$ .**p11.** What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap?**p12.** $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$ . Compute the smallest possible value of $N$ .**p13.** Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years?**p14.** Say there is $1$ rabbit on day $1$ . After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$ ?**15.** Ajit draws a picture of a regular $63$ -sided polygon, a regular $91$ -sided polygon, and a regular $105$ -sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have?**p16.** Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p17.** Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$ 's adjacent.**p18.** From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$ , where $a$ and $b$ are positive integers. Compute $a + b$ .**p19.** Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$ . He starts by putting the first marble in bucket $1$ , the second marble in bucket $2$ , the third marble in bucket $3$ , etc. After placing a marble in bucket $9$ , he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag? **p20.** What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$ ? PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).
|
1
| 47
|
|
<image>The diagram shows a cube with 6 unit edges, and the points $K$ and $L$ are the trisection points of the edge $A E$. The cube is divided into parts by the planes $L H G$ and $K F G$. What is the volume of the part containing vertex $B$?
|
138
| 48
|
|
<image>The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
|
52
| 49
|
|
<image>Problem 10.8. Rectangle $ABCD$ is such that $AD = 2AB$. Point $M$ is the midpoint of side $AD$. Inside the rectangle, there is a point $K$ such that $\angle AMK = 80^{\circ}$ and ray $KD$ is the bisector of angle $MKC$. How many degrees does angle $KDA$ measure?
|
35
| 50
|
|
<image>3. From a right prism with a square base of side length $L_{1}$ and height $H$, we extract a frustum of a pyramid, not necessarily a right one, with square bases of side lengths $L_{1}$ (for the lower base) and $L_{2}$ (for the upper base), and height $H$. The two resulting pieces are shown in the following image. If the volume of the frustum of the pyramid is 2/3 of the total volume of the prism, what is the value of $L_{1} / L_{2}$?
|
\frac{1+\sqrt{5}}{2}
| 51
|
|
<image>Problem 10.5. On the side $AB$ of the rectangle $ABCD$, a circle $\omega$ is constructed with $AB$ as its diameter. Let $P$ be the second intersection point of the segment $AC$ and the circle $\omega$. The tangent to $\omega$ at point $P$ intersects the segment $BC$ at point $K$ and passes through point $D$. Find $AD$, given that $KD=36$.
|
24
| 52
|
|
<image>7.4. What is the minimum number of kings that need to be placed on a chessboard so that they attack all unoccupied squares? (A king attacks the squares that are adjacent to its square by side or corner).
|
9
| 53
|
|
<image># 8. Variant 1. Identical coins are laid out on a table in the shape of a hexagon. If they are laid out so that the side of the hexagon consists of 2 coins, then 7 coins are enough, and if the side consists of 3 coins, then a total of 19 coins are required. How many coins are needed to build a hexagon with a side consisting of 10 coins?
|
271
| 54
|
|
<image>Rowena has a very long, level backyard. From a particular point, she determines that the angle of elevation to a large tree in her backyard is $15^{\circ}$. She moves $40 \mathrm{~m}$ closer and determines that the new angle of elevation is $30^{\circ}$. How tall, in metres, is the tree?
|
20
| 55
|
|
<image>Problem 9.8. On the side $CD$ of trapezoid $ABCD (AD \| BC)$, a point $M$ is marked. A perpendicular $AH$ is dropped from vertex $A$ to segment $BM$. It turns out that $AD = HD$. Find the length of segment $AD$, given that $BC = 16$, $CM = 8$, and $MD = 9$.
|
18
| 56
|
|
<image>Problem 8. Seryozha placed numbers from 1 to 8 in the circles so that each number, except one, was used exactly once. It turned out that the sums of the numbers on each of the five lines are equal. Which number did Seryozha not use?
|
6
| 57
|
|
<image>Problem 7.8. Given an isosceles triangle $ABC (AB = BC)$. On the ray $BA$ beyond point $A$, point $E$ is marked, and on side $BC$, point $D$ is marked. It is known that $$ \angle ADC = \angle AEC = 60^{\circ}, AD = CE = 13. $$ Find the length of segment $AE$, if $DC = 9$.
|
4
| 58
|
|
<image>B4 Simplify the expression $3^{2 n-1} \cdot 4^{n+1}+9^{n+1} \cdot 2^{2 n-1}+6^{2 n-1}$ and write it in the form of a power with base 6. (7 points) Space for solving tasks of set $B$. ## PROBLEMS FOR THIRD YEAR STUDENTS Time for solving: 90 minutes. In set A, a correct answer will be scored 2 points, while an incorrect answer will result in a deduction of half a point. Write the answers of set A in the left table.
|
6^{2n+1}
| 59
|
|
<image>Problem 5.4. The school principal, the caretaker, and the parent committee, failing to agree with each other, each bought a carpet for the school auditorium, which is $10 \times 10$. After thinking about what to do, they decided to place all three carpets as shown in the picture: the first carpet $6 \times 8$ - in one corner, the second carpet $6 \times 6$ - in the opposite corner, and the third carpet $5 \times 7$ - in one of the remaining corners (all dimensions are in meters). Find the area of the part of the hall covered by carpets in three layers (give the answer in square meters).
|
6
| 60
|
|
<image>Problem 5.3. A cuckoo clock is hanging on the wall. When a new hour begins, the cuckoo says "cuckoo" a number of times equal to the number the hour hand points to (for example, at 19:00, "cuckoo" sounds 7 times). One morning, Maxim approached the clock when it was 9:05. He started turning the minute hand until he advanced the clock by 7 hours. How many times did "cuckoo" sound during this time? <patch_newline>
|
43
| 61
|
|
<image>Problem 6.7. Anya places pebbles on the sand. First, she placed one stone, then added pebbles to form a pentagon, then made a larger outer pentagon with pebbles, then another outer pentagon, and so on, as shown in the picture. The number of stones she had arranged on the first four pictures: 1, 5, 12, and 22. If she continues to form such pictures, how many stones will be on the 10th picture? -
|
145
| 62
|
|
<image>## Task 4 - 220734 In the display window of a sports store, there is a stack of 550 balls of the same size. The stack consists of horizontal layers. Each layer contains balls arranged in a rectangular pattern, as shown in the image. The numbers $a$ and $b$ are exactly 1 less in each layer compared to the corresponding numbers in the layer below. In the bottom layer, 10 is the smaller of the two numbers $a, b$. In the top layer, 1 is the smaller of the two numbers $a, b$. Determine the number of balls in the bottom layer from these details!
|
130
| 63
|
|
<image>A group of eight students have lockers that are arranged as shown, in two rows of four lockers with one row directly on top of the other. The students are allowed to paint their lockers either blue or red according to two rules. The first rule is that there must be two blue lockers and two red lockers in each row. The second rule is that lockers in the same column must have different colours. How many ways are there for the students to paint their lockers according to the rules?
|
6
| 64
|
|
<image>Problem 11.5. Quadrilateral $ABCD$ is inscribed in a circle. It is known that $BC=CD, \angle BCA=$ $64^{\circ}, \angle ACD=70^{\circ}$. A point $O$ is marked on segment $AC$ such that $\angle ADO=32^{\circ}$. How many degrees does the angle $BOC$ measure?
|
58
| 65
|
|
<image>14. Asymmetric coin. (From 9th grade, 2 points) Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is unknown which coin is asymmetric, but it is known that the asymmetric coin lands heads with a probability of \( p = 0.6 \). Billy Bones tossed the gold coin, and heads came up immediately. Then Billy Bones started tossing the silver coin, and heads came up only on the second toss. Find the probability that the asymmetric coin is the gold one.
|
\frac{5}{9}
| 66
|
|
<image>17) The census employee on the Island of Knights and Knaves must determine the type (Knight or Knave) and the educational level of the residents (Knights always tell the truth, while Knaves always lie). In an apartment inhabited by a married couple, he only gets these responses: Husband: We are both graduates. Wife: We are both of the same type. How many boxes can the employee fill in with certainty?
|
3
| 67
|
|
<image>In the figure below, triangle \( \triangle ABC \) is right-angled at \( C \), and both \( BCDE \) and \( CAFG \) are squares. If the product of the areas of triangles \( EAB \) and \( BFA \) is 64, determine the area of triangle \( \triangle ABC \). <patch_newline>
|
8
| 68
|
|
<image>8-4. There are 50 parking spaces on a car park, numbered from 1 to 50. Currently, all parking spaces are empty. Two cars, a black one and a pink one, have arrived at the parking lot. How many ways are there to park these cars such that there is at least one empty parking space between them? If the black and pink cars switch places, this is considered a new way.
|
2352
| 69
|
|
<image>Problem 10.1. In each cell of a $5 \times 5$ table, a natural number is written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of three numbers inside any $1 \times 3$ rectangle is 23. What is the central number in the table?
|
16
| 70
|
|
<image>In the right isosceles triangle $A O B$, the points $P, Q$, and $S$ are chosen on the sides $O B, O A$, and $A B$, respectively, such that $P Q R S$ is a square. If the lengths of $O P$ and $O Q$ are $a$ and $b$, respectively, and the area of the square $P Q R S$ is $2 / 5$ of the area of the triangle $A O B$, determine the value of $a / b$.
|
2
| 71
|
|
<image>4.1. To the fraction $\frac{1}{6}$, some fraction was added, and the result turned out to be a proper fraction with a denominator less than 6. What is the largest fraction that could have been added?
|
\frac{19}{30}
| 72
|
|
<image>A1. The colors red, white, and blue are used to paint square tiles. Each tile is divided into a $2 \times 2$ grid of smaller squares, each of which can be painted one of the colors. How many differently painted tiles can you get? (Two tiles are considered the same if they can be rotated to look identical.
|
24
| 73
|
|
<image>Problem 9.6. Given an obtuse triangle $ABC$ with an obtuse angle $C$. On its sides $AB$ and $BC$, points $P$ and $Q$ are marked such that $\angle ACP = CPQ = 90^\circ$. Find the length of the segment $PQ$, if it is known that $AC = 25$, $CP = 20$, and $\angle APC = \angle A + \angle B$.
|
16
| 74
|
|
<image>Problem 9.7. Two parallel lines are drawn through points $A(0 ; 14)$ and $B(0 ; 4)$. The first line, passing through point $A$, intersects the hyperbola $y=\frac{1}{x}$ at points $K$ and $L$. The second line, passing through point $B$, intersects the hyperbola $y=\frac{1}{x}$ at points $M$ and $N$. What is $\frac{A L-A K}{B N-B M}$ ?
|
3.5
| 75
|
|
<image>Problem 9.5. Point $M$ is the midpoint of side $B C$ of triangle $A B C$, where $A B=17$, $A C=30, B C=19$. A circle is constructed with side $A B$ as its diameter. An arbitrary point $X$ is chosen on this circle. What is the minimum value that the length of segment $M X$ can take?
|
6.5
| 76
|
|
<image>Problem 9.7. In triangle $ABC$, the bisector $AL$ is drawn. Points $E$ and $D$ are marked on segments $AB$ and $BL$ respectively such that $DL = LC$, $ED \parallel AC$. Find the length of segment $ED$, given that $AE = 15$, $AC = 12$.
|
3
| 77
|
|
<image>3. (7 points) Four mole burrows $A, B, C, D$ are connected sequentially by three tunnels. Each minute, the mole runs through a tunnel to one of the adjacent burrows. In how many ways can the mole get from burrow $A$ to $C$ in 16 minutes?
|
987
| 78
|
|
<image>18. The diagram shows a track layout for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they like, returning to point $A$. The young driver, Yura, spends one minute on the path from $A$ to $B$ or back. Yura also spends one minute on the loop. The loop can only be driven counterclockwise (arrows indicate possible directions of movement). Yura does not turn back halfway and does not stop. The race duration is 10 minutes. Find the number of possible different routes (sequences of passing sections). #
|
34
| 79
|
|
<image>7. It is known that there exists a natural number $N$ such that $$ (\sqrt{3}-1)^{N}=4817152-2781184 \cdot \sqrt{3} $$ Find $N$.
|
16
| 80
|
|
<image>Problem 8.2. The numbers from 1 to 9 were placed in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other - 21. What is the sum of the numbers in the five shaded cells?
|
25
| 81
|
|
<image>In the following maze, each of the dashed segments is randomly colored either black or white. What is the probability that there will exist a path from side \( A \) to side \( B \) that does not cross any of the black lines?
|
\frac{1}{2}
| 82
|
|
<image>In the figure below, all the small squares on the grid are equal. What is the measure of the angle $\angle A E F$? Justify your answer.
|
90^\circ
| 83
|
|
<image>Problem 8.3. In triangle $ABC$, the sides $AC=14$ and $AB=6$ are known. A circle with center $O$, constructed on side $AC$ as the diameter, intersects side $BC$ at point $K$. It turns out that $\angle BAK = \angle ACB$. Find the area of triangle $BOC$.
|
21
| 84
|
|
<image>4. Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of 2, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points) Solution. Let the area of each of the given faces be \( S \). If the face lies in the plane of projection, then the projection of the tetrahedron is equal to the area of this face \( \Pi = S \). When rotated by an angle \( 0 < \varphi < 30^\circ \), the area of the projection is \( \Pi = S \cos \varphi < S \). When rotated by an angle \( 30^\circ < \varphi < 90^\circ \), the area of the projection is \[ \Pi = S \cos \varphi + S \cos \psi = S \cos \varphi + S \cos \left(\pi - \frac{\pi}{3} - \varphi\right) = S \cos \varphi + S \cos \left(\frac{2\pi}{3} - \varphi\right). \] \[ \Pi' = S \left(-\sin \varphi + \sin \left(\frac{2\pi}{3} - \varphi\right)\right), \quad \Pi' = 0 \text{ when } \varphi = \frac{\pi}{3}. \] The maximum of the function in the considered interval is achieved at \[ \varphi = \frac{\pi}{3}, \quad \Pi = 2 S \cos \left(\frac{\pi}{3}\right) = 2 S \cdot \frac{1}{2} = S. \] When rotated by an angle \( 90^\circ < \varphi < 120^\circ \), the area of the projection is \( \Pi = S \cos \left(\frac{\pi}{2} - \varphi\right) = S \sin \varphi < S \). When \( \varphi = \frac{2\pi}{3} \), the area \( \Pi = S \).
|
1
| 85
|
|
<image>365. Railway Switch. How can two trains pass each other using the switch depicted here and continue moving forward with their locomotives? The small side spur is only sufficient to accommodate either a locomotive or one car at a time. No tricks with ropes or flying are allowed. Each change of direction made by one locomotive counts as one move. What is the minimum number of moves? For a more convenient solution, draw the railway tracks on a piece of paper and place a hryvnia coin and three two-kopek coins (heads up) to represent the left train, and a hryvnia coin with two two-kopek coins (tails up) to represent the right train.
|
14
| 86
|
|
<image>B3. Veronika has a sheet of graph paper with $78 \times 78$ squares. She wants to cut the sheet into smaller pieces, each of which will have either 14 or 15 squares, with each cut dividing one piece of paper into two along one of the lines on the paper. What is the minimum number of times Veronika must cut the paper? ## Problems for 3rd Grade Time for solving: 180 minutes. Each problem in set A has exactly one correct answer. In set A, we will award two points for a correct answer and deduct one point for an incorrect answer. Write the answers for set A in the left table, leave the right table blank. The committee will consider only the answers written in the table for set A.
|
405
| 87
|
|
<image># 7.1. Condition: A carpenter took a wooden square and cut out 4 smaller equal squares from it, the area of each of which was $9 \%$ of the area of the larger one. The remaining area of the original square was $256 \mathrm{~cm}^{2}$. Find the side of the original square. Express your answer in centimeters.
|
20
| 88
|
|
<image>3. (7 points) Four mole burrows $A, B, C, D$ are connected sequentially by three tunnels. Each minute, the mole runs through a tunnel to one of the adjacent burrows. In how many ways can the mole get from burrow $A$ to $C$ in 20 minutes?
|
6765
| 89
|
|
<image>In the following drawing, $ABCD$ is a square and points $E$ and $F$ are on sides $BC$ and $CD$ such that $AEF$ is a right triangle, $AE=4$ and $EF=3$. What is the area of the square?
|
\frac{256}{17}
| 90
|
|
<image>4. Let $x$ and $y$ be the lengths of two segments on the left side. The length of the third segment is $L-(x+y)$. The sample space consists of all pairs $(x, y)$ for which $$ \left\{\begin{array} { l } { 0 \leq x \leq L } \\ { 0 \leq y \leq L } \\ { 0 \leq L - ( x + y ) \leq L } \end{array} \Leftrightarrow \left\{\begin{array}{l} 0 \leq x \leq L \\ 0 \leq y \leq L \\ 0 \leq x+y \leq L \end{array}\right.\right. $$ On the coordinate plane, they form a triangle with vertices $(0,0),(L, 0),(0, L):$ The area of this triangle is $L^{2} / 2$. To solve the problem, it is more convenient to calculate the probability of unfavorable events, which are determined by the system of inequalities $$ \left\{\begin{array} { l } { x \leq \frac { 5 } { 1 2 } L } \\ { y \leq \frac { 5 } { 1 2 } L } \\ { L - ( x + y ) \leq \frac { 5 } { 1 2 } L } \end{array} \Leftrightarrow \left\{\begin{array}{l} x \leq \frac{5}{12} L \\ y \leq \frac{5}{12} L \\ x+y \geq \frac{7}{12} L \end{array}\right.\right. $$ On the coordinate plane, they also form a right triangle with legs $L / 4$: The area of this triangle is $\frac{1}{32} L^{2}$. The probability of unfavorable events is the ratio of the areas $$ q=\frac{\frac{1}{32} L^{2}}{\frac{1}{2} L^{2}}=\frac{1}{16} $$ And the desired probability is $$ p=1-q=1-\frac{1}{16}=\frac{15}{16} $$
|
\frac{15}{16}
| 91
|
|
<image>In the diagram, the circles with centres $B$ and $D$ have radii 1 and $r$, respectively, and are tangent at $C$. The line through $A$ and $D$ passes through $B$. The line through $A$ and $S$ is tangent to the circles with centres $B$ and $D$ at $P$ and $Q$, respectively. The line through $A$ and $T$ is also tangent to both circles. Line segment $S T$ is perpendicular to $A D$ at $C$ and is tangent to both circles at $C$. (a) There is a value of $r$ for which $A S=S T=A T$. Determine this value of $r$. (b) There is a value of $r$ for which $D Q=Q P$. Determine this value of $r$. (c) A third circle, with centre $O$, passes through $A, S$ and $T$, and intersects the circle with centre $D$ at points $V$ and $W$. There is a value of $r$ for which $O V$ is perpendicular to $D V$. Determine this value of $r$.
|
2+\sqrt{5}
| 92
|
|
<image>B4 If all students in the class were to sit in their own desks, there would be 11 desks too few. If, however, two students were to sit at each desk, there would be 5 desks too many. How many desks and how many students are there in the class? (7 points) Space for solving the problems of set $B$. ## PROBLEMS FOR THE SECOND YEAR Time for solving: 90 minutes. In set A, a correct answer will be worth two points, while a wrong answer will result in a deduction of half a point. Enter the answers for set A in the left table.
|
21
| 93
|
|
<image>Problem 6. (Variant 2). $C A K D$ is a square with a side length of 6. On side $C D$, a point $B(B D=2)$ is chosen, and on line $A D$, a point $E$ is chosen such that the perimeter of triangle $B E C$ is the smallest possible. Then, on line $D C$, a point $F$ is marked such that the perimeter of triangle $F E A$ is the smallest possible. Find $E F$. ## Construction and Proof: Mark point $\mathrm{B}_{1}$ on side DK $\left(\mathrm{B}_{1} \mathrm{D}=\mathrm{BD} \Rightarrow B_{1} B \perp A D\right)$. Draw line $\mathrm{B}_{1} \mathrm{C}$, which intersects AD at point E. The perimeter of triangle CBE is the smallest because, among all possible points $\mathrm{E}_{1}$ on line $\mathrm{AD}$, the sum of the lengths of segments $\mathrm{B}_{1} \mathrm{E}+\mathrm{EC}$ is the smallest ($\mathrm{B}_{1} \mathrm{E}+\mathrm{EC}<\mathrm{B}_{1} \mathrm{E}_{1}+\mathrm{E}_{1} \mathrm{C}$ - triangle inequality) and $\mathrm{B}_{1} \mathrm{E}=\mathrm{EB}$. Similarly, mark point $\mathrm{A}_{1}$ on side $\mathrm{AC}$ $\left(\mathrm{A}_{1} \mathrm{C}=\mathrm{AC}\right)$. Draw line $\mathrm{A}_{1} \mathrm{E}$, which intersects CD at point F. The perimeter of triangle AFE is the smallest because, among all possible points $F_{1}$ on line $\mathrm{AD}$, the sum of the lengths of segments $\mathrm{A}_{1} \mathrm{~F}+\mathrm{EF}$ is the smallest $\left(\mathrm{A}_{1} \mathrm{~F}+\mathrm{EF}<\mathrm{A}_{1} \mathrm{~F}_{1}+\mathrm{F}_{1} \mathrm{E}\right.$ - triangle inequality) and $\mathrm{A}_{1} \mathrm{~F}=\mathrm{FA}$. <patch_newline>
|
0.3\sqrt{34}
| 94
|
|
<image>4. A frog jumps over lily pads. It can jump 7 lily pads to the right or 4 lily pads to the left. What is the minimum number of jumps needed for the frog to move exactly 2024 lily pads to the right?
|
297
| 95
|
|
<image>The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.
|
65
| 96
|
|
<image>In the drawing below, $C$ is the intersection point of $A E$ and $B F$, $A B=B C$, and $C E=C F$. If $\angle C E F=50^{\circ}$, determine the angle $\angle A B C$.
|
20
| 97
|
|
<image>5. (10 points) A hare jumps in one direction along a strip divided into cells. In one jump, it can move either one or two cells. In how many ways can the hare reach the 12th cell from the 1st cell? #
|
144
| 98
|
|
<image>\section*{Task 1 - 231021} On a chessboard, a queen can move such that from her position, she can reach all fields in the horizontal and vertical rows and the fields of the two diagonals intersecting at her position. In the diagram, the queen's position is marked by a black field, and the reachable fields are marked with dots. Letters and numbers at the edges are to help name the fields (here, the queen is on d2). On a \(5 \times 5\) square of fields, 5 queens are to be placed such that no queen stands on a field that can be reached by another. Determine whether this is possible, and if so, find all such placements that cannot be transformed into each other by rotation or reflection!
|
2
| 99
|
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