9000141505
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\frac{(x + 2)!}
{x!} = 2\cdot \frac{x!}
{(x - 2)!} + 3!
\]
\(\{4\}\)
\(\{4;1\}\)
\(\{1\}\)
Combinatorics
9000141506
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({11\above 0.0pt
4} \right) =\left ({11\above 0.0pt
x} \right)
\]
\(\{4;7\}\)
\(\{4\}\)
\(\{ - 4\}\)
9000141507
Level:
B
Assuming \(x,y\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({x\above 0.0pt
y}\right)^{2} - 2\cdot \left({x\above 0.0pt
y}\right) - 3 = 0
\]
\(\{[3;1];[3;2]\}\)
\(\{[3;1]\}\)
\(\{[3;1];[1;3]\}\)
9000141508
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({x\above 0.0pt
x}\right) +\left ({x + 1\above 0.0pt
x} \right) +\left ({x + 2\above 0.0pt
x} \right) +\left ({x + 3\above 0.0pt
x} \right) = \frac{x^{3} + 59}
{6}
\]
\(\{1\}\)
\(\{4\}\)
\(\{10\}\)
9000141509
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following inequality.
\[
2\cdot \left({x - 1\above 0.0pt
x - 3}\right) + x\cdot (x - 9)\leq - 8
\]
\(\{3;4;5\}\)
\(\{1;2;3;4;5\}\)
\([ 1;5] \)
9000141510
Level:
B
Assuming \(x\in \mathbb{N}\),
\(n\geq 2\), find the solution set of the following inequality.
\[
\left({ x\above 0.0pt
x - 2}\right)\cdot \left({x\above 0.0pt
2}\right) - 11\cdot \left({x\above 0.0pt
2}\right) + 28 < 0
\]
\(\{4\}\)
\(\{5;6\}\)
\((4;7)\)
1003024701
Level:
C
What is the coefficient at \( x^7 \) in the expansion of \( (1-3x)^9 \)?
\( -78\:732 \)
\( 59\:049 \)
\( -59\:049 \)
\( 78\:732 \)
1003024702
Level:
C
Find the constant term in the expansion of \( \left(2+3x^2\right)^{30} \).
\( 2^{30} \)
\( 2^{29} \)
\( 3\cdot2^{30} \)
\( 3\cdot2^{29} \)
1003024703
Level:
C
Let the term with \( x^{-3} \) in the expansion of \( \left(4x^{-\frac12}-\frac12\right)^{10} \) equals \( 105 \). Find the value of \( x \).
\( 8 \)
\( 9 \)
\( 10 \)
\( 11 \)
1003024704
Level:
C
Suppose the coefficient of the quadratic term (i.e. the coefficient at \( x^2 \)) in the expansion of \( (1+x)^n \) equals \( 300 \). Find the value of the exponent \( n \).
\( 25 \)
\( 24 \)
\( 30 \)
\( 20 \)