Math competition "Kangaroo" - 2005 - Student - Lithuania

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For which of the following values of x is the value of the expression  the smallest?


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How many numbers from 2 to 100 are equal to the cube of an integer?



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Five cards are lying on the table in the order 1, 3, 5, 4, 2. You must get the cards in the order 1, 2, 3, 4, 5. Per move, any two cards may be interchanged. How many moves do you need al least?



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If 888 • 111 = 2 • (2 • n)², and n is a positive integer, n equals:


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In the diagram every of the eight kangaroos can jump to any empty square. What is the least number of kangaroos that must jump so that each row and each column have exactly two kangaroos?



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A square piece of paper has been cut in three pieces. Two of them are in the picture on the right. What is the third one?



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The sum of four consecutive positive integers cannot be equal to:


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А 3 х 3 х 3 cube weighs 810 grams. If we drill three holes through it as shown, each of which is a 1 x 1 x 3 rectangular parallelepiped, the weight of the remaining solid is:



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If f is a function such that f(x+1)=2f(x) -2002 holds for all integer values of x and f(2005) = 2008, then f(2004) equals:


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We are given three semi-circles as shown. ABEF is a rectangle and the radius of each of the semi-circles is 2 cm. E and F are the centers of the bottom semi-circles. The area of the shaded region (in cm²) is:



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A mother kangaroo and her baby Jumpy are jumping around the stadium with a perimeter of 330 m. Both of them make 1 jump every second. The mother's jumps are 5 m long, while Jumpy's jumps are 2 m long. They both start at the same point and move in the same direction. After 25 seconds Jumpy get tired and stops while his mother continues to jump. How long is it until she is next to Jumpy again?


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Henny paints each face of several wooden cubes white or black, using both colours on each cube. How many different colourings are possible?



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The sum the dots on opposite faces of a die always equals 7. A die rolls as shown below. At the starting point (A) the top face is 3. Which will be the face at the end point (В)?



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A box contains 60 tickets: some red. some blue and some white. If all red tickets were replaced by blue tickets, then there would be twice as many blue tickets as white tickets; but if all the white tickets were replaced with blue ones, then there would be three times as many blue tickets as red tickets. The number of blue tickets in the box is:


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Let a and b be two shorter sides of the right-angled triangle. Then the sum of the diameter of the incircle and that of the circumcircle of this triangle is equal to:



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Let M be the set of all real numbers x for which the inequality holds. Then M is:


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1+2-3-4 + 5 + 6- 7- 8 +...+ 2001 + 2002 - 2003 - 2004 + 2005 =


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Two bottles of equal volume contain both juice and water. The ratios of the volume of juice to water are, respectively, 2:1 and 4:1. We put all the contents of the two bottles into one big bottle. Then the ratio of juice to water in this bottle will be:


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The diagram shows a rectangle ABEF and a triangle ABC. We know that the angle ACE equals angle CBE. If FC = 6 and CE = 2 then the area of ABC is:



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Every other day Charles always says the truth, otherwise he lies. Today he stated exactly four of the following sentences. Which one couldn't he have stated today?


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Which of the following numbers can be expressed as the product of four different integers, each of them greater than 1 ?


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In the pyramid SABC all plane angles with vertex S are equal to 90°. The areas of the lateral faces SAB, SAC and SBC are 3, 4 and 6, respectively. Find the volume of SABC.



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If the sum of the digits of m is 30, then the sum of the digits of m + 3 cannot be:


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In a bag we have 17 balls numbered by 5 +k • 125, k = 0,..., 16, i.e. by 5, 130, 255. 380, 505, ... 1755, 1880, 2005. If we select several balls at random, what is the smallest number of balls needed to guarantee that the selection contains at least one pair of balls that add tin to 2010?



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If , which of the following expressions has the value


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The positive integer m has exactly two divisors. The positive integer n has exactly five divisors. How many divisors does the number m • n have? (The unity is a divisor. The integer itself is a divisor.)


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A positive integer has к odd divisors and n even divisors. Which of the following can be the value of the quotient ? (The unity is a divisor. The integer itself is a divisor.)


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Start with a number, double it and then subtract 1. After applying this procedure 98 more times (starting each time from the previous result) you get 2100 + 1. Which was the number you started with?


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In the quadrilateral ABC D the diagonal B D is the bisector of  ABC and AC = ВС. Given  В DC = 80° and ACB = 20°,  BAD is equal to:



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Henry must travel from A to В and he plans to go at a certain speed. He would like to arrive earlier than planned and notes that travelling at a speed 5 km/h faster than planned he will arrive 5 hours earlier and travelling at a speed 10 km/h faster than planned he will arrive 8 hours earlier. What is his planned speed?


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