A

9000139701

Level: 
A
There are \(15\) athletes in an athletic meeting. Determine in how many ways it is possible to obtain the results on the first six places of the scoreboard if the place on scoreboard cannot be shared (one athlete per one place on scoreboard).
\(\frac{15!} {9!} =3\:603\:600\)
\(6^{15}=470\:184\:984\:576\)
\(15!\, 6!=941\:525\:544\:960\:000\)
\(\frac{15!} {9!\, 6!}=5\:005\)

9000139502

Level: 
A
The average mass of \(30\) eggs on a plate is \(60\, \mathrm{g}\). From this amount we remove five eggs. The total mass of these five eggs is \(280\, \mathrm{g}\). Find the change in the average mass of the remaining eggs on the plate.
The average mass of eggs increases by \(0.8\, \mathrm{g}\).
The average mass of eggs decreases by \(4\, \mathrm{g}\).
The average mass of eggs increases by \(4\, \mathrm{g}\).
The average mass of eggs increases by \(12\, \mathrm{g}\).

9000139503

Level: 
A
The average mass of the pear in a basket is \(150\, \mathrm{g}\). Find the change in the average mass of the pears in the basket if one pear has been removed from the basket.
There is not enough information to solve this problem.
The average mass of the pears increases by \(7.5\, \mathrm{g}\).
The average mass of the pears decreases by \(7.5\, \mathrm{g}\).
The average mass of the pears does not change.

9000139707

Level: 
A
A Morse code utilized dots and dashes to encode letters of an alphabet. Find the number of signals of the length from \(1\) to \(4\) which can be obtained from dots and dashes.
\(2 + 2^{2} + 2^{3} + 2^{4}=30\)
\(1 + 2 + 3! + 4!=33\)
\(\frac{4!} {3!\, 2!}=2\)
\(2 \cdot 1 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4=20\)

9000139505

Level: 
A
The average mass of twelve oranges is \(120\, \mathrm{g}\). To this amount we add another six oranges with the average mass \(150\, \mathrm{g}\). Find the change in the average mass of oranges.
The average mass increases by \(10\, \mathrm{g}\).
The average mass increases by \(8.3\, \mathrm{g}\).
The average mass increases by \(25\, \mathrm{g}\).
The average mass decreases by \(8.3\, \mathrm{g}\).

9000139708

Level: 
A
The shelf contains \(15\) books. From this amount, \(9\) books are in English and \(6\) books in other languages. Find the number of possibilities how to rearrange the books on the shelf, if all English books have to be on the left and the other on the right.
\(9!\, 6!=261\:273\:600\)
\(9^{6}=531\:441\)
\(\frac{9!} {6!}=504\)
\(\frac{9!} {6!\, 3!}=84\)

9000139507

Level: 
A
The average mass of five melons is \(2\: 400\, \mathrm{g}\). We have to add another melon such that the new average value of all six melons will be \(2\: 420\, \mathrm{g}\). Find the mass of the sixth melon.
\(2\: 520\, \mathrm{g}\)
\(2\: 540\, \mathrm{g}\)
\(2\: 480\, \mathrm{g}\)
\(2\: 460\, \mathrm{g}\)

9000139501

Level: 
A
Ten apples in a box have average mass \(200\, \mathrm{g}\). We remove one apple of the mass \(200\, \mathrm{g}\) from the box. What is the change in the average mass of the apples from the box?
The average mass of the apples does not change.
The average mass of the apples decreases by \(20\, \mathrm{g}\).
The average mass of the apples increases by \(20\, \mathrm{g}\).
There is not enough information to solve this problem.

9000140002

Level: 
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\). \[ \frac{x+a} {a} = ax - 1\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{-1;1\} & \emptyset \\ a\notin\{-1;0;1\} & \left\{\frac{2a}{(a-1)(a+1)}\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-1 & \emptyset \\ a\notin\{-1;0\} & \left\{\frac{2a}{(a-1)(a+1)}\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{-1;1\} & \mathbb{R} \\ a\notin\{-1;0;1\} & \left\{\frac{2a}{(a-1)(a+1)}\right\} \\\hline \end{array}\)