Math competition "Kangaroo" - 2004 - Student - Lithuania

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If m pens are bought at n euros each, and n pens at m euros each (m  n), then the average cost per pen, in euros, is:

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A pyramid has 17 faces. How many vertices does it have?

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The smallest real number satisfying the ineguality x² - 2004  0? 

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Each Martian has one, two, or three tentacles on its head. Exactly 1 % of the Martian popu-lation consists of individuals with three tentacles, exactly 97% comprise Martians with two tentacles, and the remaining 2% consists of individuals with one tentacle. What percent of Martians have more tentacles on their head than the average of the whole Martian popula-tion?

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In a square of side i, where s is an odd integer, the squares of side 1 on the diagonals are colored (like in the picture, where the square is of side 7). How many white squares are there?


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How many two-digit numbers exist whose square and cube end in the same digit?

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Square ABC D consists of 18 smaller squares, 17 of which have sides equal to 1. The area of the square ABCD is:

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How many right triangles can be formed by joining three vertices of a given regular 14-gon?

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This is a multiplication table. What two letters represent the same number?
 

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On the circumference of radius r three points X, Y and A are marked such that XY = r,XY  AY(see the figure). How many degrees has the angle XAY ? 


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In the plane Oxy, how many squares with vertex A(— 1; —1) exist such that at least one of the coordinate axes is an axis of symmetry of the sąuare?

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There are 100 cards in a non-transparent envelope, numbered with integers from 1 to 100. There is a different number on each card. What is the smallest number of cards we have to puli out of the envelope at random to be sure that the product of the numbers on the chosen cards is divisible by 4?

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The set of all pairs (x, y) which satisfy conditions xy < 0 ir x² + y² = 4 is on the graph:


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In the figure the two equilateral triangles ABC and ECD have sides of length 2 and 1 respectively. The area of the quadrilateral АBСЕ is:

 

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How many positive integers can be written as a₀ +  a₁ * 3 + a₂ * 3²+ а₃  * 3³ +a₄ * 3⁴,  if a₀ , a₁, a₂, a₃, a₄ belong to the set {—1,0, 1}? 

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The number (  -  )²  is 

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How many vertices are there in a regular polygon the sum of whose interior angles is one seventh of that of a regular 16-gon?

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A circle K is inscribed in a quarter circle with radius 6 as shown in the figure. What is the radius of circle K?

 

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For a geometrie sequence (а_n) the following ineąualities hold: a₃ < a₂ < a₄. Tada 

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What is the second digit from the right of the number 11²⁰⁰⁴ ? 

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An election was held in Herbville. Every voter who voted for the Broccoli Party had already eaten broccoli. Of the remaining voters who voted for other parties 90% had never eaten broccoli. What percent did the Broccoli Party get in the election if precisely 46% of all voters in the election had eaten broccoli?

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A parallelogram is divided into 4 triangles as shown in the figure. Of the following possibilities for the areas of the triangles at most one can be true. Which one is it?
 

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The figure shows graphs of funetions f and g defined on real numbers. Each graph consists of two perpendicular halflines. Which equality is satisfied for every real number x?


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An equilateral triangle ABC with sides of length 4 is given. The radius of the circular are, with center at A, which divides the triangle into two parts of equal area is:

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A game starts with a sequence of two hundred zeroes. In the first round we add 1 to every number. In the second round we add 1 to the second number and to every second number after it. In the third round we add 1 to the third number and to every third number after it, and so on. What number is in the 120th position after two hundred rounds?

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How many triangles can be drawn with vertices in the 18 points shown in the figure?
 


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If the sum of all the numbers that can be formed by permutation of the three different digits 0
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Taškų skaičius už teisingą atsakymą: 5      

Klausimas #28

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The number m = 999... 9 consists of 999 nines. What is the sum of the digits of m² ?

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Taškų skaičius už teisingą atsakymą: 5      

Klausimas #29

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sin⁸ 75° - cos⁸ 75° is equal to:

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Taškų skaičius už teisingą atsakymą: 5      

Klausimas #30

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Let ABC D be a convex quadrilateral with an area of 1 where AB and BD are the bases of two isosceles triangles ADB and BCD respectively (as shown).
 
The product AC • BD is equal to:

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Taškų skaičius už teisingą atsakymą: 5