Math competition "Kangaroo" - 2009 - Junior - Lithuania

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Which of these numbers is a multiple of 3?


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What is the smallest number of bold points in the figure one needs to remove so that no 3 of the remaining points were collinear?



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2009 people participated in a popular race. The number of people John has outstripped was three times larger than the number of people who won over John. In what place has John been ranked in the race?


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What is the value of the of 1000?


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A long sequence of digits has been composed by writing the number 2009 repeatedly 2009 times. The sum of those odd digits in the sequence that are immediately followed by an even digit is equal to


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The picture shows a solid formed with 6 triangular faces. At each vertex there is a number. For each face we consider the sum of the 3 numbers at the vertices of that face. If all the sums are the same and two of the numbers are 1 and 5, as shown, what is the sum of all the 5 numbers?



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How many positive integers have as many digits in the decimal representation of their square as of their cube?


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The area of the triangle in the picture is 80 m2 and the radius of the circles centered at the vertices is 2 m. What is the measure, in m2, of the shaded area?



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Leonard has written a sequence of numbers so that each number (from the third number in the sequence) was a sum of the previous two numbers in the sequence. The fourth number in the sequence was 6 and the sixth number in the sequence was 15. What was the seventh number in the sequence?



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 A triangle has an angle of 68°. Bisectors of the other two angles are drawn. How many degrees is the angle with the question sign?



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At each test, the mark can be 0, 1, 2, 3, 4, or 5. After 4 tests, Mary's average was 4. One of the sentences cannot be true. Which is it?


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 The Borromean rings have a surprising property: the three of them cannot be separated without destroying them, but once one of them is removed (regardless which one), the other two are not linked anymore. Which of the following figures shows the Borromean rings?


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25 people are standing in a queue on the island of nobles and liars. Everyone, except the first person in the queue, said that the person ahead of him in the queue was a liar, and the first man in the queue said that all the people standing behind him were liars. How many liars were there in the queue? (Nobles always speak the truth, and liars always tell lies.)



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 If a • b = ab + a + b, and 3 Δ 5 = 2Δx  then x equal is to:


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Around the vertices of a square circles are drawn: 2 large and 2 small ones. The large circles are tangent to each other and to both the small circles.

What is the ratio between the radius of a large circle and that of a small circle?


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The difference between and 10 is less than 1. How many such integer n exist?


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Man Friday wrote down in a row several different natural numbers smaller than 11. Robinson Crusoe examined these numbers and noticed with satisfaction that in each pair of the neighboring numbers one of the numbers was divisible by another. How many numbers at most could Man Friday write down?


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3 circular hoops are joined together so that they intersect at the right angles as shown. A ladybird lands on an intersection and crawls as follows: it travels along a quarter-circle, turns to the right 90°, then travels along another quarter-circle and turns to the left 90°.

Proceeding in this way, how many quarter-circles will it travel along before she returns again to her starting point?


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How many zeros should be inscribed instead of * in the^ decimal fraction 1. * 1 in order to a get a number that is smaller than , but larger than ?


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If a = 2²⁵, b = 8⁸ and с = 3¹¹, then


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How many ten-digit numbers, composed only of digits 1, 2 and 3, do there exist, in which any two neighboring digits differ by 1?


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A young kangaroo has 2009 unit lxlxl cubes that he has used in forming a cuboid. He also has 2009 stickers lxi that he must use to color the outer surface of the cuboid. The kangaroo has achieved his goal and some stickers were left. How many stickers were left?


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Bob wants to place draughts into cells of the 4 x 4 board so that the numbers of the draughts in any row and any column were different (more than one draught can be placed into one cell as well as the cell can be empty). What is the smallest possible number of the draughts placed on the board?



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Some oranges, peaches, apples, and bananas were put in a row so that somewhere in the row each type of fruit can be found side by side with each other type of fruit. What is the least number of fruits in the row?


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What is the least integer n, for which (2² - 1) • (3² - 1) • (4² - 1) •... • (n² - 1) is a perfect square?


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All the divisors of the number N, unequal to N and to 1, were written in turn. It occurred that the greatest of the divisors in the line is 45 times greater than the smallest one. How many numbers satisfy this condition?


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 A kangaroo is sitting in the origin of a coordinate system. It can jump 1 unit vertically or horizontally. How many points are there in the plane at which the kangaroo can be after 10 jumps?


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Let AD be a median in the triangle ABC. The angle ACB is 30°, the angle ADB is 45°.

What is the measure of the angle BAD?


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Find the minimal quantity of numbers one should remove from the set {1, 2, 3, 16} so that the sum of any 2 remaining numbers were not a perfect square.


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A prime number is defined as being strange if it is either a one-digit prime or if it has two or more digits, but both numbers obtained by omitting either its first or last digit are also strange. How many strange primes are there?


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