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Problem set 4. Part I
{{exercise_number}}. What is the \latex{ 2005 }th digit in the decimal form of \latex{\frac{1}{7}} after the decimal point?
{{exercise_number}}. How tall is the tower whose shadow is \latex{ 15\,m } when the shadow of a vertically positioned \latex{ 1\,m } long cane is \latex{ 75\,cm }?
{{exercise_number}}. Find the set of numbers that can be written in the place of * in \latex{\frac{2}{*}-\frac{*}{5}=\frac{1}{15}.}
{{exercise_number}}. In a junction traffic is directed by traffic lights positioned at the right hand side of each lane; the direction of each lane is indicated by the arrows. What is the colour shown on each light when light \latex{ 1 } is red? (Every light is either red or green, the number of green lights is maximal such that there are no two crossing lanes with both green lights.)
1
2
6
5
3
4
{{exercise_number}}. If we double both the base and the exponent in \latex{a^{b}}, we get the number \latex{ z }. Determine \latex{ x } for which \latex{a^{b}} times \latex{x^{b}} equals \latex{ z }.
{{exercise_number}}. The average of five numbers is \latex{ 12 }. If we add \latex{ 10 } to each number, then multiply the resulting numbers by \latex{ 4 } and finally subtract \latex{ 10 } from each, we get five new numbers. Determine the average of these numbers.
{{exercise_number}}. The tank of a car is \latex{\frac{1}{4}} full, then \latex{ 18 } litres of petrol is refueled. Now the tank is \latex{ 0.7 } full. Determine the capacity of the tank in litres.
{{exercise_number}}. Determine the length indicated by the thick line if \latex{ A } and\latex{ B } are the centres of the circles and \latex{AB = 12\,cm.}
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
{{exercise_number}}. Teams of \latex{ 3 } students from six schools are competing in cross-country running. Every student has the same chance of winning (or being \latex{ 2 }nd or being \latex{ 3 }rd). What is the probability that the first three places are occupied by students from the same school?
{{exercise_number}}. For swimming \latex{ 3\,km }, a swimmer either has to complete the length of the pool \latex{ 90 } times or its perimeter \latex{ 25 } times. What is the area of the surface of the water (the area of the pool in \latex{m^{2}}) ?
{{exercise_number}}. Determine the sum of the digits in the decimal form of the number \latex{(10^{4n^{2}+8 }+1 ),n\in \N.}
{{exercise_number}}. We pick a \latex{ 3 }-digit positive integer \latex{ N } at random. What is the probability that \latex{\log _{2}N} is an integer?
Problem set 4. Part II/A
{{exercise_number}}. A merchant buys an item for \latex{ 900 } EUR.
- What price tag should he apply if he wants to have \latex{ 20 }% profit even after giving \latex{ 10 }% discount?
- What will the price of the item be after three consecutive discounts of \latex{ 10 }% each?
{{exercise_number}}.
- Solve the following equation: \latex{\sqrt[5]{x^{2} }-\sqrt[5]{x}=2.}
- Solve the following system of equations: \latex{\begin{rcases}6x+6y=12 \\ 3x+3y=6xy\end{rcases}}
{{exercise_number}}. Three brothers inherit the quadrilateral shaped field seen on the figure. There are roads \latex{ AE, AF, AC, CG } and \latex{ CH } through the field, where the points \latex{ E, F, G, H } are the trisecting points of the corresponding sides.
- Is it possible for the brothers to share the land equally such that the borders of the parts are along the roads, the parts belonging to one owner do not have to be adjacent but the total area of land owned by each brother is the same?
- How many different fair partitions exist if there are no borders other than the roads?
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{ H }
{{exercise_number}}.
- The sum of the first \latex{ 50 } elements of an arithmetic progression is \latex{ 200 }, the sum of the next \latex{ 50 } elements is \latex{ 270 }. Find the first element of the progression.
- A fruit pressing machine squeezes one quarter of the juice from a carrot for the first press, then for every consecutive press it squeezes one quarter of the remaining juice. How many times does it need to press to obtain at least \latex{\frac{2}{3}} of the juice?
Problem set 4. Part II/B
{{exercise_number}}.
- The average result of a class on a test was \latex{ 80 } points. The average of the girls was \latex{ 83 } points while the average of the boys was \latex{ 71 } points. Determine the percentage of girls in the class.
- In the test consisting of \latex{ 25 } questions, a correct answer is worth \latex{ 5 } points, an incorrect answer is worth \latex{ 0 } point and not answering is worth \latex{ 1 } point. A few students were asked about their result. They replied as follows: Andrew – \latex{ 127 } points, Barnaby – \latex{ 124 } points, Christopher – \latex{ 121 } points, Dennis – \latex{ 118 } points, Edward – \latex{ 115 } points. Are these statements possible? Who is mistaken for sure?
{{exercise_number}}. Compute the mass of water on Earth if its volume is \latex{1.3875\times 10^{15}\,m^{3},} the density of salt water is \latex{1,035\,kg/m^{3},} the density of freshwater is \latex{1,000\,kg/m^{3}} and the percentage distribution of waters is shown in the diagram (density is the quotient of mass and volume).
Pacific Ocean
(\latex{ 51.37 }%)
(\latex{ 51.37 }%)
Atlantic Ocean
(\latex{ 25.2 }%)
(\latex{ 25.2 }%)
freshwater
Indian Ocean
(\latex{ 20.72 }%)
(\latex{ 20.72 }%)
(\latex{ 2.71 }%)
{{exercise_number}}. For the sake of the security of the Eiffel Tower, its profile is described by the following logarithmic function:
\latex{y=91\times \left|\ln \frac{|x|}{62.5} \right|.}
The first level of the tower is \latex{ 57.63\,m }, the second level is \latex{ 115.75\,m } and the third one is \latex{ 276.13\,m } above ground.
- Find the width of the base of the tower.
- Determine the width of the balcony on the second level.
- How far can we see from the balcony on the third level? (Find the distance of the horizon from the tower in bee line considering the Earth as a sphere with a radius of \latex{6,37\times 10^{6}\,m}.)
\latex{ y }
\latex{ x }
