1003108801
Level:
B
Express the repeating decimal \( 2.\overline{4} \) as a fraction in lowest terms.
\( \frac{22}9 \)
\( \frac49 \)
\( \frac{22}7 \)
\( \frac{12}5 \)
\( \frac9{22} \)
Infinite series
1003108802
Level:
B
Express the repeating decimal \( 0.5\overline{83} \) as a fraction in lowest terms.
\( \frac{289}{495} \)
\( \frac{83}{990} \)
\( \frac{495}{289} \)
\( \frac{298}{495} \)
\( \frac{583}{1\,000} \)
1003108803
Level:
B
What is the sum of the repeating numbers \( 0.\overline{27} \) and \( 0.4\overline{63} \)?
\( \frac{81}{110} \)
\( \frac{110}{81} \)
\( \frac{80}{111} \)
\( \frac{79}{110} \)
\( \frac{81}{100} \)
1003108804
Level:
B
What is the expression \( \frac{0.4\overline6}{0.\overline{63}} \) equal to?
\( \frac{11}{15} \)
\( \frac{15}{11} \)
\( \frac{11}{14} \)
\( \frac{14}{11} \)
\( \frac{13}{15} \)
1003108805
Level:
B
We are given the equation
\[ \sum\limits_{n=0}^{\infty}\left(\frac2x\right)^n=\frac{4x-3}{3x-4} \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \{ 6 \} \)
\( \{ 1 \} \)
\( \{ 6;1 \} \)
\( \{\ \} \)
\( \{-6;-1\} \)
1003108806
Level:
B
We are given the equation
\[ \sum\limits_{n=1}^{\infty}(x+2)^{2n} =\frac13 \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \left\{ -\frac32;-\frac52 \right\} \)
\( \left\{ -\frac32\right\} \)
\( \left\{ -\frac52 \right\} \)
\( \left\{\ \right\} \)
\( \left\{ \frac32;\frac52 \right\} \)
1003108807
Level:
B
We are given the equation
\[\log x+\log\sqrt x+\log\sqrt[4]x+\log\sqrt[8]x+\dots=2\]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \{ 10 \} \)
\( \{ 1 \} \)
\( \{ 100 \} \)
\( \{ \ \} \)
\( \left\{ \frac1{10} \right\} \)
1003108808
Level:
B
We are given the equation
\[ \sum\limits_{n=1}^{\infty}2^{nx}=1 \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \{ -1 \} \)
\( \{ 1 \} \)
\( \{ 1;-1 \} \)
\( \{\ \} \)
\( \{ 0 \} \)
1003108809
Level:
B
We are given the equation
\[ \sum\limits_{n=1}^{\infty} (\sin x)^{2n-2}=2\cdot\,\mathrm{tg}\,x \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac14\pi+k\cdot\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac34\pi+k\cdot\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac34\pi+k\cdot\frac{\pi}2\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac18\pi+k\cdot\frac{\pi}2\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac14\pi+k\cdot\frac{\pi}2\right\} \)
2010006903
Level:
B
What is the sum of the repeating numbers \( 0.\overline{45} \) and \( 0.2\overline{81} \)?
\( \frac{81}{110} \)
\( \frac{59}{110} \)
\( \frac{110}{81} \)
\( \frac{36}{110} \)
\( \frac{81}{100} \)