1003030106
Level:
A
How many solutions does the following equation have?
\[ 2^x\cdot5^x=100^{3-x} \]
Exactly one solution
No solutions
Infinitely many solutions
Exactly two solutions
A
1003030105
Level:
A
How many solutions does the following equation have?
\[ \sqrt{81^{x+1}}=3^{2x}\]
No solutions
Infinitely many solutions
Exactly one solution
Exactly two solutions
1003030104
Level:
A
Decide which of the following intervals contains \( d \), if \( \left(0.1^{d-1}\right)^3=10^{d-1} \).
\( [-1;3] \)
\( [9;13] \)
\( [4;8] \)
\( [-6;-2] \)
1003030103
Level:
A
Solve.
\[ 8^{2x}=16\sqrt[3]4\]
\( x=\frac79 \)
\( x=\frac{11}{12} \)
\( x=\frac49 \)
\( x=\frac13 \)
1003030102
Level:
A
Solve.
\[ 5^{8-2x}=1\]
\( x=4 \)
\( x=-4 \)
\( x=\frac72 \)
The equation has no solution.
1003030101
Level:
A
Solve.
\[ 2^x=-4\]
The equation has no solution.
\( x=2 \)
\( x=-2 \)
\( x=-\frac12 \)
1003037306
Level:
A
The ratio of two more than five times \( x \) and five is equal to the sum of \( x \) and \( a \). Determine \( a \).
\( 0.4 \)
It is not possible to determine the number \( a \) uniquely without knowing the number \( x \).
\( 2 \)
\( 10 \)
1003037305
Level:
A
Suppose we increase the numerator of five-ninth by an integer and we decrease the denominator of five-ninth by the same integer. We get the fraction with numerator six times greater than the denominator. Find the integer.
\( 7 \)
\( 3 \)
There is no such an integer.
\( 1 \)
1003037304
Level:
A
The product of two consecutive natural numbers is the same as the sum of the square of the smaller one and eight. Find the smaller number.
\( 8 \)
\( 4 \)
\( 2 \)
There is not such a natural number.
1003037303
Level:
A
Two more than eight times of the unknown number is the same as one-third of the difference of one and the number. Find the number.
\( -\frac15 \)
\( -\frac7{23} \)
\( -\frac5{27} \)
\( -\frac7{21} \)