C

2010001406

Level: 
C
All \(96\) children in the summer camp were to choose the chocolate champion - between milk chocolate and dark chocolate. At the end milk chocolate got \(30\) more votes than dark chocolate. \(26\) children couldn’t decide though and gave their votes to both chocolates. How many votes did dark chocolate get?
\(46\)
\( 76\)
\( 26\)
\( 82\)

2010001306

Level: 
C
Factor the following polynomial. \[ x^{8} - 1 \]
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + 1\right )\left (x^{4} + 1\right )\)
\(\left (x - 1\right )^2\left (x + 1\right )^2\left (x^{2} + 1\right )^2\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{3} + 1\right )^2\)
\(\left (x - 1\right )^2\left (x + 1\right )^2\left (x^{4} + 1\right )\)

2010000904

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{2} {3}\right \}\), find the quotient of the polynomials: \[ (x^{2} - x - 1) : (3x + 2) \]
\(\frac{1} {3}x -\frac{5} {9} + \frac{\frac{1} {9} } {3x+2}\)
\(\frac{1} {3}x -\frac{5} {9} - \frac{\frac{19} {9} } {3x+2}\)
\(\frac{1} {3}x -\frac{1} {9} + \frac{\frac{7} {9} } {3x+2}\)
\(\frac{1} {3}x -\frac{1} {9} - \frac{\frac{11} {9} } {3x+2}\)

2010000903

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{\pm 1\right \}\), find the quotient of the polynomials: \[ (-3x^{4} + 2x^{2} -4) : (x^{2} + 1) \]
\(- 3x^{2} + 5 - \frac{9} {x^{2}+1}\)
\(- 3x^{2} - 5 - \frac{9} {x^{2}+1}\)
\(- 3x^{2} + 5 +\frac{1} {x^{2}+1}\)
\(- 3x^{2} - 5 +\frac{1} {x^{2}+1}\)

2010000805

Level: 
C
In a parallel circuit the total resistance \( R \) of three components with the resistances \( R_1 \), \( R_2 \), \( R_3 \) is given by the formula \( \frac1R=\frac1{R_1}+\frac1{R_2}+\frac1{R_3} \). Express \( R_2 \) from this formula.
\( R_2=\frac{RR_1R_3}{R_1R_3-R(R_1+R_3)} \)
\( R_2=\frac{R-R_1-R_3}{RR_1R_3} \)
\( R_2=R-\frac{R_1R_3}{R_1+R_3} \)
\( R_2=\frac{R_1R_3-R(R_1+R_3)}{RR_1R_3} \)