Found problems: 15
IV Soros Olympiad 1997 - 98 (Russia), grade8
[b]p1.[/b] What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$ and $15\%$ solutions?
[b]p2.[/b] Which number is greater: $199,719,971,997^2$ or $199,719,971,996 * 19,9719,971,998$ ?
[b]p3.[/b] Is there a convex $1998$-gon whose angles are all integer degrees?
[b]p4.[/b] Is there a ten-digit number divisible by $11$ that uses all the digits from$ 0$ to $9$?
[b]p5.[/b] There are $20$ numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is $0$.
[b]p6.[/b] Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than $180$ degrees?
[b]p7.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon.
[b]p8.[/b] Give an example of a natural number that is divisible by $30$ and has exactly $105$ different natural factors, including $1$ and the number itself.
[b]p9.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes
$5 * 8 + 7 + 1 = 48$
$2 * 2 * 6 = 24$
$5* 6 = 30$
a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued?
b) What does the number$ 9$ mean among the Antipodes?
Clarifications:
a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems.
b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system?
[b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts?
PS.1. There was typo in problem $9$, it asks for $2^3$ and not $23$.
PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
V Soros Olympiad 1998 - 99 (Russia), grade7
[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img]
[b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.)
[b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle?
[b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$.
[b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
V Soros Olympiad 1998 - 99 (Russia), grade8
[b]p1.[/b] Given two irreducible fractions. The denominator of the first fraction is $4$, the denominator of the second fraction is $6$. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction?
[b]p2.[/b] Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio $1: 4$. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio $1:3$, on the third -$ 1:1$. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race?
[b]p3.[/b] A quadrilateral is inscribed in a circle, such that the center of the circle, point $O$, is lies inside it. Let $K$, $L$, $M$, $N$ be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles $\angle KOM$ and $\angle LOC$ are perpendicular (Fig.).
[img]https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png[/img]
[b]p4.[/b] Prove that the number$$\underbrace{33...33}_{1999 \,\,\,3s}1$$ is not divisible by $7$.
[b]p5.[/b] In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^o$ with it. Find the greatest angle of this triangle.
[b]p6.[/b] Given a $7\times 8$ rectangle made up of 1x1 cells. Cut it into figures consisting of $1\times 1$ cells, so that each figure consists of no more than $5$ cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
VI Soros Olympiad 1999 - 2000 (Russia), grade8
[b]p1.[/b] Can a number ending in $1999$ be the square of a natural number?
[b]p2.[/b] The Three-Headed Snake Gorynych celebrated his birthday. His heads took turns feasting on birthday cakes and ate two identical cakes in $15$ minutes. It is known that each head ate as much time as it would take the other two to eat the same pie together. In how many minutes would the three heads of the Serpent Gorynych eat one pie together?
[b]p3.[/b] Find the sum of the coefficients of the polynomial obtained after opening the brackets and bringing similar terms into the expression:
a) $(7x - 6)^4 - 1$
b) $(7x - 6)^{1999}-1$
[b]p4.[/b] The general wants to arrange seven anti-aircraft installations so that among any three of them there are two installations, the distance between which is exactly $10$ kilometers. Help the general solve this problem.
[b]p5.[/b] Gulliver, whose height is $999$ millimeters, is building a tower of cubes. The first cube has a height of $1/2$ a lilikilometer, the second - $1/4$ a lilikilometer, the third - $1/8$ a lilikilometer, etc. How many cubes will be in the tower when its height exceeds Gulliver's height. ($1$ lilikilometer is equal to $1000$ lilimeters).
[b]p6.[/b] It is known that in any pentagon you can choose three diagonals from which you can form a triangle. Is there a pentagon in which such diagonals can be chosen in a unique way?
[b]p7.[/b] It is known that for natural numbers $a$ and $b$ the equality $19a = 99b$ holds. Can $a + b$ be a prime number?
[b]p8.[/b] Vitya thought of $5$ integers and told Vanya all their pairwise sums:
$$0, 1, 5, 7, 11, 12, 18, 24, 25, 29.$$
Help Vanya guess the numbers he has in mind.
[b]p9.[/b] In a $3 \times 3$ square, numbers are arranged so that the sum of the numbers in each row, in each column and on each major diagonal is equal to $0$. It is known that the sum of the squares of the numbers in the top row is $n$. What can be the sum of the squares of the numbers in the bottom line?
[b]p10.[/b] $N$ points are marked on a circle. Two players play this game: the first player connects two of these points with a chord, from the end of which the second player draws a chord to one of the remaining points so as not to intersect the already drawn chord. Then the first player makes the same “move” - draws a new chord from the end of the second chord to one of the remaining points so that it does not intersect any of the already drawn ones. The one who cannot make such a “move” loses. Who wins when played correctly? (A chord is a segment whose ends lie on a given circle)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here[/url].
V Soros Olympiad 1998 - 99 (Russia), grade7
[b]p1.[/b] Ivan Ivanovich came to the store with $20$ rubles. The store sold brooms for $1$ ruble. $17$ kopecks and basins for $1$ rub. $66$ kopecks (there are no other products left in the store). How many brooms and how many basins does he need to buy in order to spend as much money as possible? (Note: $1$ ruble = $100$ kopecks)
[b]p2.[/b] On the road from city A to city B there are kilometer posts. On each pillar, on one side, the distance to city A is written, and on the other, to B. In the morning, a tourist passed by a pillar on which one number was twice the size of the other. After walking another $10$ km, the tourist saw a post on which the numbers differed exactly three times. What is the distance from A to B? List all possibilities.
[b]p3.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365 days on the next New Year's Eve?
[b]p4.[/b] What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from $100$ to $999$?
[b]p5.[/b] An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to the denominator of the third fraction, the result was $13/23$. What fraction was written on the board?
[b]p6.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property?
[b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?
[b]p8.[/b] The square is divided by straight lines into $25$ rectangles (fig. 1). The areas of some of them are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img]
[b]p9.[/b] Petya multiplied all natural numbers from $1$ to his age inclusive. The result is a number
$$8 \,\, 841 \,\,761993 \,\,739 \,\,701954 \,\,543 \,\,616 \,\,000 \,\,000.$$ How old is Petya?
[b]p10.[/b] There are $100$ integers written in a line, and the sum of any three in a row is equal to $10$ or $11$. The first number is equal to one. What could the last number be? List all possibilities.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
V Soros Olympiad 1998 - 99 (Russia), grade8
[b]p1.[/b] Two proper ordinary fractions are given. The first has a numerator that is $5$ less than the denominator, and the second has a numerator that is $1998$ less than the denominator. Can their sum have a numerator greater than its denominator?
[b]p2.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in $365$ days on the next New Year's Eve?
[b]p3.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property?
[b]p4.[/b] In the quadrilateral $ABCD$, the extensions of opposite sides $AB$ and $CD$ intersect at an angle of $20^o$; the extensions of opposite sides $BC$ and $AD$ also intersect at an angle of $20^o$. Prove that two angles in this quadrilateral are equal and the other two differ by $40^o$.
[b]p5.[/b] Given two positive integers $a$ and $b$. Prove that $a^ab^b\ge a^ab^a.$
[b]p6.[/b] The square is divided by straight lines into $25$ rectangles (fig.). The areas of some of They are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img]
[b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $ 17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?
[b]p8.[/b] In expression $$(a-b+c)(d+e+f)(g-h-k)(\ell +m- n)(p + q)$$ opened the brackets. How many members will there be? How many of them will be preceded by a minus sign?
[b]p9.[/b] In some countries they decided to hold popular elections of the government. Two-thirds of voters in this country are urban and one-third are rural. The President must propose for approval a draft government of $100$ people. It is known that the same percentage of urban (rural) residents will vote for the project as there are people from the city (rural) in the proposed project. What is the smallest number of city residents that must be included in the draft government so that more than half of the voters vote for it?
[b]p10.[/b] Vasya and Petya play such a game on a $10 \times 10 board$. Vasya has many squares the size of one cell, Petya has many corners of three cells (fig.). They are walking one by one - first Vasya puts his square on the board, then Petya puts his corner, then Vasya puts another square, etc. (You cannot place pieces on top of others.) The one who cannot make the next move loses. Vasya claims that he can always win, no matter how hard Petya tries. Is Vasya right?
[img]https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
IV Soros Olympiad 1997 - 98 (Russia), grade8
[b]p1.[/b] a) There are barrels weighing $1, 2, 3, 4, ..., 19, 20$ pounds. Is it possible to distribute them equally (by weight) into three trucks?
b) The same question for barrels weighing $1, 2, 3, 4, ..., 9, 10$ pounds.
[b]p2.[/b] There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now?
[b]p3.[/b] What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\% $of the value of $x$?
[b]p4.[/b] Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”).
[b]p5.[/b] There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure?
[img]https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png[/img]
[b]p6.[/b] The natural number $a$ is less than the natural number $b$. In this case, the sum of the digits of number $a$ is $100$ less than the sum of the digits of number $b$. Prove that between the numbers $ a$ and $b$ there is a number whose sum of digits is $43$ more than the sum of the digits of $a$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
IV Soros Olympiad 1997 - 98 (Russia), grade6
[b]p1.[/b] The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles?
[b]p2.[/b] From point $O$ on the plane there are four rays $OA$, $OB$, $OC$ and $OD$ (not necessarily in that order). It is known that $\angle AOB =40^o$, $\angle BOC = 70^o$, $\angle COD = 80^o$. What values can the angle between rays $OA$ and $OD$ take? (The angle between the rays is from $0^o$ to $180^o$.)
[b]p3.[/b] Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles.
[b]p4.[/b] Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year?
[b]p5.[/b] The difference between two four-digit numbers is $7$. How much can the sums of their digits differ?
[b]p6.[/b] The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written on the board. In one move you can increase any of the numbers by $3$ or $5$. What is the minimum number of moves you need to make for all the numbers to become equal?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
VI Soros Olympiad 1999 - 2000 (Russia), grade7
[b]p1.[/b] Cities A, B, C, D and E are located next to each other along the highway at a distance of $5$ km from each other. The bus runs along the highway from city A to city E and back. The bus consumes $20$ liters of gasoline for every $100$ kilometers. In which city will a bus run out of gas if it initially had $150$ liters of gasoline in its tank?
[b]p2.[/b] Find the minimum four-digit number whose product of all digits is $729$. Explain your answer.
[b]p3.[/b] At the parade, soldiers are lined up in two lines of equal length, and in the first line the distance between adjacent soldiers is $ 20\%$ greater than in the second (there is the same distance between adjacent soldiers in the same line). How many soldiers are in the first rank if there are $85$ soldiers in the second rank?
[b]p4.[/b] It is known about three numbers that the sum of any two of them is not less than twice the third number, and the sum of all three is equal to $300$. Find all triplets of such (not necessarily integer) numbers.
[b]p5.[/b] The tourist fills two tanks of water using two hoses. $2.9$ liters of water flow out per minute from the first hose, $8.7$ liters from the second. At that moment, when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if the capacity of the smaller one is $12.5$ liters?
[b]p6.[/b] Is it possible to mark 6 points on a plane and connect them with non-intersecting segments (with ends at these points) so that exactly four segments come out of each point?
[b]p7.[/b] Petya wrote all the natural numbers from $1$ to $1000$ and circled those that are represented as the difference of the squares of two integers. Among the circled numbers, which numbers are more even or odd?
[b]p8.[/b] On a sheet of checkered paper, draw a circle of maximum radius that intersects the grid lines only at the nodes. Explain your answer.
[b]p9.[/b] Along the railway there are kilometer posts at a distance of $1$ km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is $14$ km. What is the maximum distance between the red pillars?
[b]p10.[/b] The island nation is located on $100$ islands connected by bridges, with some islands also connected to the mainland by a bridge. It is known that from each island you can travel to each (possibly through other islands). In order to improve traffic safety, one-way traffic was introduced on all bridges. It turned out that from each island you can leave only one bridge and that from at least one of the islands you can go to the mainland. Prove that from each island you can get to the mainland, and along a single route.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
V Soros Olympiad 1998 - 99 (Russia), 8.5
Points $A$, $B$ and $C$ lie on one side of the angle with the vertex at point $O$, and points $A'$, $B'$ and $C'$ lie on the other. It is known that$ B$ is the midpoint of the segment $AC$, $B'$ is the midpoint of the segment $A'C'$, and lines $AA'$, $BB'$ and $CC'$ are parallel (fig.). Prove that the centers of the circles circumscribed around the triangles $OAC$, $OA'C$ and $OBB'$ lie on the same straight line.
[img]https://cdn.artofproblemsolving.com/attachments/d/6/92831077781bc45f25e9f71077034f84753a59.png[/img]
IV Soros Olympiad 1997 - 98 (Russia), grade6
[b]p1.[/b] For $25$ bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost?
[b]p2.[/b] Cut the square into the figure into$ 4$ parts of the same shape and size so that each part contains exactly one shaded square. [img]https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png[/img]
[b]p3.[/b] The numerator and denominator of the fraction are positive numbers. The numerator is increased by $1$, and the denominator is increased by $10$. Can this increase the fraction?
[b]p4.[/b] The brother left the house $5$ minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister?
[b]p5.[/b] Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.)
[b]p6.[/b] Give an example of a natural number divisible by $6$ and having exactly $15$ different natural divisors (counting $1$ and the number itself).
[b]p7.[/b] In a round dance, $30$ children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance?
[b]p8.[/b] A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got $4$ holes. The positions of three of them are marked in figure Where might the fourth hole be? [img]https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png[/img]
[b]p9 [/b] The numbers 1$, 2, 3, 4, 5, _, 2000$ are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining $1000 $ numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number?
[b]p10.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis in $1996$ jumps if he must not get to points with coordinates divisible by $4$ (points $0$, $\pm 4$, $\pm 8$ etc.)?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
IV Soros Olympiad 1997 - 98 (Russia), grade7
[b]p1.[/b] The oil pipeline passes by three villages $A$, $B$, $C$. In the first village, $30\%$ of the initial amount of oil is drained, in the second - $40\%$ of the amount that will reach village $B$, and in the third - $50\%$ of the amount that will reach village $C$ What percentage of the initial amount of oil reaches the end of the pipeline?
[b]p2.[/b] There are several ordinary irreducible fractions (not necessarily proper) with natural numerators and denominators (and the denominators are greater than $1$). The product of all fractions is equal to $10$. All numerators and denominators are increased by $1$. Can the product of the resulting fractions be greater than $10$?
[b]p3.[/b] The garland consists of $10$ light bulbs connected in series. Exactly one of the light bulbs has burned out, but it is not known which one. There is a suitable light bulb available to replace a burnt out one. To unscrew a light bulb, you need $10$ seconds, to screw it in - also $10$ seconds (the time for other actions can be neglected). Is it possible to be guaranteed to find a burnt out light bulb:
a) in $10$ minutes,
b) in $5$ minutes?
[b]p4.[/b] When fast and slow athletes run across the stadium in one direction, the fast one overtakes the slow one every $15$ minutes, and when they run towards each other, they meet once every $5$ minutes. How many times is the speed of a fast runner greater than the speed of a slow runner?
[b]p5.[/b] Petya was $35$ minutes late for school. Then he decided to run to the kiosk for ice cream. But when he returned, the second lesson had already begun. He immediately ran for ice cream a second time and was gone for the same amount of time. When he returned, it turned out that he was late again, and he had to wait $50$ minutes before the start of the fourth lesson. How long does it take to run from school to the ice cream stand and back if each lesson, including recess after it, lasts $55$ minutes?
[b]p6.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon.
[b]p7.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes
$5 * 8 + 7 + 1 = 48$
$2 * 2 * 6 = 24$
$5* 6 = 30$
a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued?
b) What does the number 9 mean among the Antipodes?
Clarifications:
a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems.
b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system?
[b]p8.[/b] They wrote the numbers $1, 2, 3, 4, ..., 1996, 1997$ in a row. Which digits were used more when writing these numbers - ones or twos? How long?
[b]p9.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis $in 1996$ jumps if he must not get to points with coordinates divisible by $ 4$ (points $0$, $\pm 4$, $\pm 8$, etc.)?
[b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
IV Soros Olympiad 1997 - 98 (Russia), grade7
[b]p1.[/b] In the correct identity $(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)$ two numbers were replaced with dots. What were these numbers?
[b]p2.[/b] A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\%$ of the amount currently available, on the other - $20\%$, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B?
[img]https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png[/img]
[b]p3.[/b] Find the angle between the hour and minute hands at $7$ hours $38$ minutes.
[b]p4.[/b] The lottery game is played as follows. A random number from $1$ to $1000$ is selected. If it is divisible by $2$, they pay a ruble, if it is divisible by $10$ - two rubles, by $12$ - four rubles, by $20$ - eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options.
[b]p5.[/b]The sum of the digits of a positive integer $x$ is equal to $n$. Prove that between $x$ and $10x$ you can find an integer whose sum of digits is $ n + 5$.
[b]p6.[/b] $9$ people took part in the campaign, which lasted $12$ days. There were $3$ people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all $12$ days they were able to appoint three people on duty, taking into account this requirement. Could this be so?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
V Soros Olympiad 1998 - 99 (Russia), 8.1 - 8.4
[b]p1.[/b] Is it possible to write $5$ different fractions that add up to $1$, such that their numerators are equal to one and their denominators are natural numbers?
[b]p2.[/b] The following is known about two numbers $x$ and $y$:
if $x\ge 0$, then $y = 1 -x$;
if $y\le 1$, then $x = 1 + y$;
if $x\le 1$, then $x = |1 + y|$.
Find $x$ and $y$.
[b]p3.[/b] Five people living in different cities received a salary, some more, others less ($143$, $233$, $313$, $410$ and $413$ rubles). Each of them can send money to the other by mail. In this case, the post office takes $10\%$ of the amount of money sent for the transfer (in order to receive $100$ rubles, you need to send $10\%$ more, that is, $110$ rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method?
[b]p4.[/b] a) List three different natural numbers $m$, $n$ and $k$ for which $m! = n! \cdot k!$ .
b) Is it possible to come up with $1999$ such triplets?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
V Soros Olympiad 1998 - 99 (Russia), grade7
[b]p1.[/b] Due to the crisis, the salaries of the company's employees decreased by $1/5$. By what percentage should it be increased in order for it to reach its previous value?
[b]p2.[/b] Can the sum of six different positive numbers equal their product?
[b]p3.[/b] Points$ A, B, C$ and $B$ are marked on the straight line. It is known that $AC = a$ and $BP = b$. What is the distance between the midpoints of segments $AB$ and $CB$? List all possibilities.
[b]p4.[/b] Find the last three digits of $625^{19} + 376^{99}$.
[b]p5.[/b] Citizens of five different countries sit at the round table (there may be several representatives from one country). It is known that for any two countries (out of the given five) there will be citizens of these countries sitting next to each other. What is the smallest number of people that can sit at the table?
[b]p6.[/b] Can any rectangle be cut into $1999$ pieces, from which you can form a square?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]