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To prove that the sum of the numbers of the exact square cannot be equal to 5. | number_theory |
( b) Will the statement of the previous challenge remain true if Petia and Wasa originally planned four natural numbers? | number_theory |
The quadratic three-member graph with the coefficients has two points with the coordinates. Show that if the distance between them is the whole number, then the fraction is parallel to the abscisse axis. | polynoms |
Can you draw on the surface of Rubik's cube a closed path that passes through each square exactly once (the top of the square does not pass through)? | combinatorics |
Dima, who came from Vrunlandia, said that there were several lakes connected by rivers, and there were three rivers coming out of each lake, and there were four rivers in each lake, so prove him wrong. | graphs |
If the number is the whole, the number is the whole. | number_theory |
Can n sit 2n + 1 round table so that no two sit next to each other more than once if (a) n = 5; (b) n = 4; (c) n is an arbitrary natural number? | graphs |
Natural numbers a, b, c, d are such that ab = cd. Prove that there are such natural u, v, w, z that a = uv, b = wz, c = uw, d = vz. | number_theory |
The ABC angle triangle is placed in the circle of. The tangential to, taken through points B and C, crosses the tangent to, taken through point A, to points K and L respectively. The straight to K parallel to AB intersects with a line drawn through L parallel to the AC at point P. Prove that BP = CP. | geometry |
The dan convex ABCDEFGH, where all the inner angles are equal to each other, and the sides are equal through one – AB = CD = EF = GH, BC = DE = FG = HA (to be called a semi-right angle). By conducting diagonals AD, BE, CF, DG, EH, FA, GB and HC. Among the parts into which these diagonals break the inner area of the ocu... | invariant |
Send comment Decision Page: < < 2 3 4 5 6 7 8 >> [Total tasks: 223] 1 2 5 10 20 50 100 with decisions | geometry |
The circulatory legs are located in the knots of an infinite leaf of celled paper whose cells are squared on the side of 1. Permitted without changing the circulatory solution by turning it around one of the legs to move the second foot into another knot on the sheet. Is it possible to change the circulatory legs in a ... | number_theory |
The secret service has N agents – 001, 002,..., 007,... n. The first agent is looking after the second, the second one is looking after the third, etc., n-i is looking after the first one. Prove that n is an odd number. | number_theory |
When the hockey tournament ended (in one lap), it turned out that each team could find a team (maybe from the same group) that scored an odd number of points in the games with the team's teams. Prove that an even number of teams participated in the tournament. | number_theory |
How many different ways are there to divide the number of 2004 into natural confluences, which are approximately equal? The numbers may be one or more. The numbers are called roughly equal if their differences are not greater than 1. The methods that differ only in order of composition are considered to be the same. | combinatorics |
On the right side of the street, there are houses with even natural numbers, on the left, with odd natural numbers, numbers increasing from the beginning of the street to the end on each side, but not necessarily running in a row. For each house on the right side of the street, there was a difference between the number... | number_theory |
At what natural n for each whole k ≥ n will there be a multiple n number with the sum of k? Send a comment. | number_theory |
The pedestrian walked six streets of the same city, every two times, but couldn't get around them every single time. | graphs |
Let the natural numerals m1, m2,..., mn are interchangeably simple. Prove that if the numbers x1, x2, xn run the full subtractive systems for the M1, m2,..., mn, respectively, the number x = x1m2...mn + m1x2m3...mn + + + m1m2...mn-1xn runs the complete subtractive system for the M1m2 module...mn. | number_theory |
From the cell cell square 100×100 was cut across the cell boundaries of 1950 dominos (two-cell rectangles). Prove that the remaining part can be cut across the cell boundaries of a four-cell figure of the species T, possibly wrapped. (If such a figure already exists among the remaining parts, it is believed that it has... | invariant |
Prove that if ABCD is a written quadrilateral, the sum of the radius of the circles in the ABC and ACD triangles is equal to the sum of the radius of the circles in the BCD and BDA triangles. Send a comment Decision Page: << 30 31 32 33 35 35 >> [Total tasks: 223] 1 2 5 10 20 50 100 with decisions | geometry |
The correct 1981 coal has 64 vertices. Prove that there is a trapeze with vertices at the points noted. | dirichlet |
Let p be a prime number greater than 2, and m/n = 1 + 1 / 2 + 1 / 3 +... + 1/p -1. | number_theory |
Is there such a number of n : that numbers a) n – 96, n + 96; b) n – 1996, n, n + 1996 simple? (All simple numbers count positive.) Send a comment Decision Task 98406 Themes: [paints] [various tasks in cutting] [Quality and equilibrium] [Tables and tournaments (other)] Complexity: 3 Klass: 7.8.9 Author: Shapovals A.V. ... | number_theory |
Let's call natural numbers similar if they're recorded with the same set of numbers (e.g. for a set of numbers 1, 1, 2 are similar to the numbers 112, 121, 211). Prove that there are three similar 1995 figures that have no zeros, that the sum of two of them is equal to the third. | number_theory |
Is it possible to form in space a closed chain of 61 identical harmonized rotating gears in such a way that the angles between the clutched gears are not less than 150 degrees? In this way: for simplicity, the six are considered to be circles; the sixs are coupled if the corresponding circles at the point of contact ha... | number_theory |
Each of the ribs of a complete graph with nine vertices painted blue or red. Prove that there are four vertices, all ribs between which are blue, or there are three vertices, all ribs between which are red. | dirichlet |
(a) The electrical circuit has a type of grid 3×3: all in a diagram of 16 knots (grace squares) that are connected by wires (parts of the grid squares). Perhaps part of the wires are burned. For one measurement, you can choose any pair of circuit knots and check whether there is a current between them (i.e., check whet... | combinatorics |
The convention brought together scientists, among whom there are friends, and it turns out that every two of them, having equal numbers of friends at the convention, have no mutual friends, to prove that there is a scientist who has exactly one friend among the participants in the convention. | graphs |
There are 101 vertebrates in each vertebrate. The number of incoming and outgoing ribs is 40. Prove that from each vertice it is possible to reach any other one, with no more than three ribs. Send a comment. Mission 35677 Themes: [Chachmat Painting] [Dirichle (Other) Principle] [Changing Composition] The difficulty: 4C... | dirichlet |
At what value of a multi-member P(x) = x1000 + ax2 + 9 is divided by x + 1? | polynoms |
Of these, 22 were held by the hand of the boy and 30 were held by the hand of the girl. | invariant |
At what whole n number n4 + 4 is composite? | number_theory |
Prove inequality | number_theory |
At what are the n values, all the coefficients of Newton's binoma degradation (a + b)n are odd? | combinatorics |
The secret service has N agents – 001, 002,..., 007,... n. The first agent is looking after the second, the second one is looking after the third, etc., n-i is looking after the first one. Prove that n is an odd number. | combinatorics |
(c) | combinatorics |
Malvina asked Buratino to write out all the nine-digit numbers made up of different numbers. Buratino forgot how the number 7 was written, so only recorded the nine-digit numbers that did not. Then Malvina suggested that he should delete the remaining three-digit number from each number, so that the remaining three-dig... | number_theory |
Prove that if the square equation with the whole coefficients has the root u = [a;], the second root will be the number | polynoms |
Prove that at k ≥ 1, the equation is: = [aFk; aFk – 1,..., aF0] where {Fk} is the sequence of Fibonacci numbers. | number_theory |
Prove that the two adjacent numbers Fbonacci Fn-1 and Fn (n ≥ 1) are mutually simple. | number_theory |
It is known that an – bn is divided by n (a, b, n – natural numbers, a ~ b). Prove what is divided by n. | number_theory |
Prove inequalities for natural n: | number_theory |
Prove that the number (m, n ≥ 0) is whole. | combinatorics |
There are several thin matches of the same length on the table. Is it always possible to paint their ends (a) in 2, (b) in three colours so that the two ends of each match are different colours and every two related matches are the same color? | dirichlet |
Each of the 102 pupils in one school is familiar with at least 68 others; prove that there are four of them who have the same number of acquaintances. | number_theory |
The natural numbers a1, a2,..., an are such that each number does not exceed its number (k≤ k) and the sum of all numbers is an even number. To prove that one of the sums a1 ± a2 ±... ± an is zero. | number_theory |
a, b, c is whole numbers; a and b are different from zero. Prove that the equation ax + by = c has solutions in whole numbers when and only when c is divided by d = SPLM(a, b). | dirichlet |
What is the lowest sum of numbers that can have the number of species 3n2 + n + 1 with natural n? | number_theory |
At the camera music festival, six musicians gathered at each concert, and some musicians perform and the others listen to them from the hall. For what is the lowest number of concerts each of the six musicians will be able to listen to (from the hall) all the others? | dirichlet |
There is an infinite arithmetical progress of natural numbers with a non-zero difference. Each of its members has a square root removed and, if there is a single number, rounded to the nearest whole. Is it possible that all the roundings were in the same direction? | dirichlet |
The cell strip 1×1000000 is broken into 100 segments. Each cell has an entire number, and each cell has a number that matches the number in the same segment. Each cell has a chip. Then they do this operation: all the chips are moved at the same time, each of them to the number of cells to the right that is shown in its... | combinatorics |
There are 1955 points. What is the maximum number of threes that can be selected so that every two threes have exactly one common point? | combinatorics |
In square table 4×4 cells, there are + and – signs as shown in the figure. It is allowed to change the sign simultaneously in all cells in the same row, in the same column or in a straight line parallel to a diagonal (in particular, it is possible to change the sign in any corner cell). Prove that no matter how many of... | invariant |
Natural numbers m1,..., mn are mutually simple in pairs. Prove that the number x = (m2...mn)\(m1) is the solution of the system x \1 (mod m1), x \0 (mod m2),... x \0 (mod mn). | number_theory |
Are there 19 such pairs of different natural numbers with the same sum of numbers that amount to 1999? | number_theory |
Set the equation x2 to 5y2 = 1 in whole numbers. | number_theory |
The teacher recorded two natural numbers on the board. The Lion multiplied the first number by the sum of the second and received 20131313. The Feday multiplied the second number by the sum of the first numerals and received the second number by the sum of the first numerals and received the second number by the sum of... | number_theory |
Each letter of the original message was replaced by a two-digit number in the Russian alphabet according to the table: | number_theory |
There are 100 bills of two types: a and b rubles, and a b (mod 101). | dirichlet |
In the city, 57 bus routes are known to be: (1) from each stop to any other stop, without a transplant; (2) for each pair of routes, there will be only one stop at which one of these routes can be moved to another; (3) for each route, there will be at least three stops. How many stops each of the 57 routes has? | combinatorics |
By using the Ledjandra formula (see task 60553), prove that the number is whole. | number_theory |
Tom and Jerome were found on the road with a pack of 11 pieces, drinking 3 cups of tea in Tom's tea shop, eating 4 pebbles and 5 bagels, drinking 9 cups of tea, eating 1 pebble and 4 bagels, and having a cup of tea, a cup of tea and a bagel with a whole number of rubles, and it turns out that Tom can pay 11 chunks with... | number_theory |
For which n is there such a closed non-self-reconclusive breakage of n links that each straight line containing one of the links of this fracture contains at least one more link? | number_theory |
Prove that there are infinitely many of these three numbers n – 1, n, n + 1, that: (a) n is represented as the sum of two squares of natural (full positive) numbers and n – 1 and n + 1 is not; (b) each of the three numbers is represented as the sum of two squares of natural numbers. | number_theory |
The sequence of natural numbers a1, a2,..., an,... so that for each n the equation an+2x2 + an+1x + an = 0 has a valid root. Can the number of members of this sequence be a) equal to 10; b) infinite? | polynoms |
M is the centre of the circle of the MD radius. Find the angular value of the arc of the circle between the sides of BA and BC if BAC = 65o. Send a comment. Decision Task 52605 Themes: [The angle between the two chords and the two sections] [Straight related to the circles (other)] Complexity: 3Klass: 8.9 Inside the ci... | geometry |
Natural numbers x and y are so that 2x2 is 1 = y15. Prove that if x > 1, x is divided by 5. | number_theory |
Is there an infinite number of such threes of whole numbers x, y, z that x2 + y2 + z2 = x3 + y3 + z3? | number_theory |
The seller and buyer have a total of 1999 rubles of coins and bills of 1, 5, 10, 50, 100, 500 and 1,000 rubles. The cat in the bag is worth a whole number of rubles, and the buyer has enough money. | number_theory |
Send a comment Decision Task 64350 Themes: [Written and described circles] [Four points on the same circle] [Subsidiary Equal Triangles] [Radial Axis] [Direct Homothetics (Other)]] The difficulty: 4+Classes: 9.10 Author: Bogdanov I.I. On the sides of the ABC sharp triangle outside it are CAKL and CBM squares. Direct N ... | geometry |
Can each whole number be recorded as the sum of the cubes of several whole numbers, which are not the same? | number_theory |
Prove that the angle between the tangent and the chord taken through the point of contact is equal to half the angle value of the arc between them. Send a comment. Send a comment. sent a comment. Decision. 55389 Theme:: [the angle between the angle and the chord] The difficulty: 3Klasss: 8.9 The factor in point A to th... | geometry |
Dana is an unbroken non-self-reconclusive broken line of 37 links. Each link has a straight line. What is the lowest number of different lines that could have worked? | combinatorics |
The graph shows the graphs of three square members. Can these numbers be selected a, b and c so that these are the graphs of three members ax2 + bx + c, bx2 + cx + a and cx2 + ax + ax + ax + b? | polynoms |
The distance between the centres of the circles is greater than the sum of their radius. Prove that the mid-points of the four common tangents of these circles lie on one line. | geometry |
(a) 21 coins of tail up on the table. One operation allows any 20 coins to be turned over. Can all coins be placed in an eagle up in a few operations? (b) The same question if 20 coins and 19 coins are allowed to be rotated. | number_theory |
Find an area where the bases are equal to 10 and 26, and the diagonal are perpendicular to the side sides. Send a comment Decision Task 67207 Themes: [The four points on the same circle] [The four points on the same circle help solve the task] [Symmetry helps solve the task] The difficulty: 4-Class: 4-Classss: 8.9.10 A... | geometry |
The light panel consists of several lamps, each of which may be in two conditions (burn or not). On the bullet, several buttons, each of which simultaneously changes the condition of a set of lamps (for each button its own), the lamps do not start to burn. (a) Prove that the number of different patterns that can be obt... | combinatorics |
The Rogue and Hoov equity rate is raised or lowered by n% every day at 12:00, where n is a fixed natural number, less than 100 (the rate is not rounded). Is there a n for which the share rate can take the same value twice? | number_theory |
Twenty-five coins are arranged at random into two groups. Then any of the existing groups are split into two groups again, and so on until each group consists of a single coin. When each group is split into two, a piece of coins is recorded in two groups. What is the sum of all the numbers recorded? | invariant |
Is there a 2016-digit number that can be converted to 2016 different 2016-digit full squares? | dirichlet |
(a) There is a piece of cheese. It is allowed to select any positive (perhaps not whole) number a × 1, and cut this piece in relation to 1 : a by weight, then cut in the same way any of the existing pieces, and so on. Is it possible to act in such a way that, after the final number of cuts, the whole cheese can be divi... | number_theory |
Find the last two digits of the decimal number 1! + 2! +... + 2001! + 2002! | number_theory |
In some kingdoms, an unlimited number of coins in n1, n2, n3,... copéeks where n1 < n < 2 < n3 <... is an infinite sequence of natural numbers. Prove that this sequence can be cut off, i.e. there will be so many Ns that any sum that can be paid without the coins issued can actually be paid only by n1, n2, nN copieks. | number_theory |
From a glass of milk, three spoons of contents are pumped into a cup of tea and sloppy, then scrape three spoons of the mixture and then pour them back into a glass of milk. What's more, tea in a cup of milk or milk in a cup of tea? | invariant |
(b) l(0) + l(1)x + l(2)x2 +... = (1 – x)-1 (1 – x3) -1 (1 – x5) -1...; | combinatorics |
There were two decks on the table, 36 cards each, the first deck was dragged and placed on a second deck. Then for each card the first deck calculated the number of cards between it and the same card of the second deck (i.e. how many cards between seven worms, between the ladies of the peak, etc.). What is the sum of t... | invariant |
(a) Two identical sixs with 14 teeth each. They were placed on each other so that the teeth match. (So the projection on the plane looks like one.) Then four pairs of matching teeth drank. Is it always possible to turn these sixs relative to each other so that the projection on the plane looks like one whole six? (Shes... | number_theory |
At the National Basketball Association, 30 teams, each of which spends 82 matches a year with other teams in the regular championship. Can the Association's management share teams (not necessarily equally) at the Eastern and Western Conferences and schedule the games so that teams from the various conferences match exa... | graphs |
There are n > 1 cities in the country, some pairs of cities are connected by two long-haul flights, while there is only one airway between each two cities (perhaps with transplants). The mayor of each city of X has calculated the number of such numberings of all cities from 1 to n, which on any airway starting in X is ... | combinatorics |
A cellular rectangular grid m×n is tied from single length ropes. Two make moves in turn. In one move, you can cut (in the middle) a single rope not previously cut. If there are no closed ropes left, the player who made the last move is considered to be the loser. Which player wins in the right game and how does he hav... | graphs |
Let P(x) and Q(x) be multi-member, with Q(x) not equal to zero. Prove that there are so many T(x) and R(x) members that P(x) = Q(x)T(x) + R(x) and DEG R(x) < degQ(x) and that T(x) and R(x) are clearly defined. | polynoms |
(b)................................................................. | combinatorics |
On board the airliner 2n passengers and the airline loaded for them n meals with chicken and n meals with fish. It is known that a passenger with a probability of 0.5 prefers chicken and a probability of 0.5 prefers fish. Call the passenger unhappy if he/she does not have what he/she prefers. (a) Find the most likely n... | number_theory |
Resolve equation in natural numbers | number_theory |
The plan of the city is a plane divided into the same right triangles. The sides of the triangles are motorways and the tops of the triangles are intersections. From points A and B on the same road (a side triangle), two cars move simultaneously in the same direction at the same speed. Once they reach any intersection,... | invariant |
The listed circle of the ABC triangle refers to the sides of BC, CA, AB in points A', B', C', respectively. Direct AA', BB' and CC' intersects at point G. The described circle of the GA'B' triangle crosses again the straight AC and BC in points CA and CB. Similarly, the points AB, AC, BC, BA are defined. Prove that the... | geometry |
In the cinema, seven rows of 10 seats each, a group of 50 children went to the morning session and then to the evening session, prove that there were two children who sat in the morning session and sat in the evening row. | dirichlet |
The book "The magic for teapots" says: Replace the letters in the word ZEMLETRATION with the same numbers and the different letters with the different ones. If the number is simple, there will be a real earthquake. Is it possible to create an earthquake in this way? | number_theory |
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