C

1103266406

Level: 
C
The medieval builder has a $5$-ell-long iron belt. His task is to shape the belt into a frame of the Romanesque window (that is the union of a rectangle and a semicircle, see the picture). Find the optimal width $x$ of the window to get as much light coming through the window as possible (i. e. the area of the window should be as large as possible). Express the result rounded in inches ($1\,\mathrm{ell} = 45\,\mathrm{inches}$).
$63$
$140$
$32$
$112$
$83$
$20$

1103266405

Level: 
C
Adam's House ($A$) is located at the distance of $0.9\,\mathrm{km}$ from the road. There is a bus stop ($B$) on this road at the distance of $1.5\,\mathrm{km}$ from the house (see the picture). Adam has overslept and needs to get to the bus stop as quickly as possible. At what distance $x$ from the nearest point $P$ should Adam reach the road knowing that he can move at the speed of $6\,\mathrm{km}/\mathrm{h}$ in rough terrain while being on the road he can move at the speed of $10\,\mathrm{km}/\mathrm{h}$?
$0.675\,\mathrm{km}$
$0.525\,\mathrm{km}$
$0.625\,\mathrm{km}$
$0.575\,\mathrm{km}$

1103266403

Level: 
C
We want to create a rabbit cage in the shape of a rectangle with sides $a$ and $b$. The cage will be divided by parallel walls into four sections with the same area (see the picture). Find the dimensions $a$ and $b$ providing we have $50\,\mathrm{m}$ of fencing wire available and we want the total area to be as large as possible. (Fencing wire will also be used for the walls.)
$a=5\,\mathrm{m}$, $b=12.5\,\mathrm{m}$
$a=4\,\mathrm{m}$, $b=15\,\mathrm{m}$
$a=4.5\,\mathrm{m}$, $b=13.75\,\mathrm{m}$
$a=6.5\,\mathrm{m}$, $b=8.75\,\mathrm{m}$

1003266402

Level: 
C
The price of an Archery game experience program for groups up to $8$ participants is $12$ EUR/person. In case of a larger group (number of participants higher than $8$), each additional person reduces the price for all participants by $0.5$ $\mathrm{EUR}$/person. Find the number of participants that will bring the organizing company maximum income and calculate the total income.
There will be a maximum income of $128$ $\mathrm{EUR}$ for $16$ participants.
There will be a maximum income of $128$ $\mathrm{EUR}$ for $8$ participants.
There will be a maximum income of $192$ $\mathrm{EUR}$ for $16$ participants.
There will be a maximum income of $192$ $\mathrm{EUR}$ for $12$ participants.
None of the answers is correct.

1103266401

Level: 
C
A producer of sterilized canned vegetables needs to reduce the production costs of a $0.5$ liter cylindrical can. Find such radius $r$ and the height $h$ of the can (in centimeters) so that its surface (i.e. the amount of material needed) is minimal.
$r\doteq 4.3\,\mathrm{cm}$, $h\doteq 8.6\,\mathrm{cm}$
$r\doteq 3.4\,\mathrm{cm}$, $h\doteq 13.8\,\mathrm{cm}$
$r\doteq 5.4\,\mathrm{cm}$, $h\doteq 5.5\,\mathrm{cm}$
$r\doteq 3.4\,\mathrm{cm}$, $h\doteq 8.6\,\mathrm{cm}$

1003047808

Level: 
C
A heavy smoker decides to reduce his daily consumption by $2$ cigarettes for $30$ days from the beginning of the next year. Then reduces his consumption again by $2$ cigarettes per day every $30$ days. How much will he save in $360$ days, given that one box of cigarettes ($20$ cigarettes) costs $80\ \mathrm{CZK}$?
$18\,720\ \mathrm{CZK}$
$624\ \mathrm{CZK}$
$4\,680\ \mathrm{CZK}$
$12\,480\ \mathrm{CZK}$
$6\,240\ \mathrm{CZK}$

1003047807

Level: 
C
At a dorm, students play with rolls of toilet paper, building a pyramid of it. There is one roll at the top of the pyramid, in each lower row there is one roll more than in the one above it. How high will the pyramid be, given the fact that they have a total of $171$ rolls and one roll is $9.5\,\mathrm{cm}$ high?
$171\,\mathrm{cm}$
$2\,\mathrm{m}$
$180\,\mathrm{cm}$
$95\,\mathrm{cm}$
$123.5\,\mathrm{cm}$

1003047806

Level: 
C
Every school must pay a registration fee for each participant it sends to a math competition. The fee for the first participant is $10$ euros, for each additional one the fee is one euro less. The maximum number of participants any school can register is $10$. Find the relationship between the price ($c$) paid by the school and the number of students registered ($n$).
$c=\frac n2(21-n)$
$c=10-\frac{n^2}2$
$c=\frac{11n}2$
$c=\frac n2(10+10n)$
$c=\frac n2(11-n)$

1003047805

Level: 
C
A cyclist plans to travel $1666\,\mathrm{km}$ in $14$ days during her vacation. She knows that the number of kilometers she rides every day will decrease by the same number so she planned her route accordingly. At the beginning of the last day she was only $80\,\mathrm{km}$ away from her goal. What is the difference between the numbers of kilometers she rode on two consecutive days?
$6\,\mathrm{km}$
$7\,\mathrm{km}$
$5\,\mathrm{km}$
$4\,\mathrm{km}$
$3\,\mathrm{km}$