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grade8

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(2)

grade 8 problems (V Soros Olympiad 1998-99 Round 1)

Source:

5/21/2024
p1. Two proper ordinary fractions are given. The first has a numerator that is 55 less than the denominator, and the second has a numerator that is 19981998 less than the denominator. Can their sum have a numerator greater than its denominator?
p2. 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 365365 days on the next New Year's Eve?
p3. The number xx is such that 15%15\% of it and 33%33\% of it are positive integers. What is the smallest number xx (not necessarily an integer!) with this property?
p4. In the quadrilateral ABCDABCD, the extensions of opposite sides ABAB and CDCD intersect at an angle of 20o20^o; the extensions of opposite sides BCBC and ADAD also intersect at an angle of 20o20^o. Prove that two angles in this quadrilateral are equal and the other two differ by 40o40^o.
p5. Given two positive integers aa and bb. Prove that aabb≥aaba.a^ab^b\ge a^ab^a.
p6. The square is divided by straight lines into 2525 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. https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png
p7. A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly 17o 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?
p8. In expression (a−b+c)(d+e+f)(g−h−k)(ℓ+m−n)(p+q)(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?
p9. 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 100100 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?
p10. Vasya and Petya play such a game on a 10×10board10 \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? https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
algebrageometrycombinatoricsnumber theorySoros Olympiad
grade 8 problems (V Soros Olympiad 1998-99 Round 2)

Source:

5/21/2024
p1. Given two irreducible fractions. The denominator of the first fraction is 44, the denominator of the second fraction is 66. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction?
p2. 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 1:41: 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 1:31:3, on the third -1:1 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?
p3. A quadrilateral is inscribed in a circle, such that the center of the circle, point OO, is lies inside it. Let KK, LL, MM, NN be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles ∠KOM\angle KOM and ∠LOC\angle LOC are perpendicular (Fig.). https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png
p4. Prove that the number33...33⏟1999   3s1\underbrace{33...33}_{1999 \,\,\,3s}1 is not divisible by 77.
p5. In triangle ABCABC, the median drawn from vertex AA to side BCBC is four times smaller than side ABAB and forms an angle of 60o60^o with it. Find the greatest angle of this triangle.
p6. Given a 7×87\times 8 rectangle made up of 1x1 cells. Cut it into figures consisting of 1×11\times 1 cells, so that each figure consists of no more than 55 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.
algebrageometrycombinatoricsnumber theorySoros Olympiad