Geometric sequences

1003170603

Level: 
C
Between the roots of the equation \( 9x^2+130x-75=0 \) insert two numbers so that the roots and the new numbers form \( 4 \) consecutive terms of a geometric sequence. What is the smaller of the two inserted numbers?
\( -\frac53 \)
\( \frac59 \)
\( -\frac59 \)
\( \frac53 \)
\( -5 \)

1003170602

Level: 
C
The lengths of a cuboid edges form \( 3 \) consecutive terms of a geometric sequence. The volume of the cuboid is \( 140\,608\,\mathrm{cm}^3 \), the sum of its shortest and its longest edge is \( 221\,\mathrm{cm} \). Find the length of its shortest edge.
\( 13\,\mathrm{cm} \)
\( 52\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
\( 208\,\mathrm{cm} \)
\( 0.25\,\mathrm{cm} \)

1003170601

Level: 
C
How many numbers do we need to insert between the numbers \( 6 \) and \( 1\,458 \) so that the inserted numbers with the given two numbers are consecutive terms of a geometric sequence? The sum of all numbers inserted must be \( 720 \).
\( 4 \)
\( 6 \)
\( 3 \)
\( 5 \)
\( 7 \)

1003109207

Level: 
C
In \( 2015 \), \( 8\,688 \) choir members sang in the largest gospel choir in Manila. Imagine, the conductor wrote an e-mail on January \( 1 \)st to three choir members. Each of them forwarded the e-mail to other three choir members on the next day and so on... On which day would all members receive the e-mail?
\( 8 \) January
\( 15 \) January
\( 2 \) February
\( 8 \) February
\( 12 \) January

1003109206

Level: 
C
In \( 2016 \), the population of the Czech Republic increased by \( 25\,000 \) to \( 10\,578\,820 \) inhabitants. How many inhabitants will there live in the Czech Republic at the end of the year \( 2026 \), if the percentage increase is the same every year?
\( 10\,832\,100 \)
\( 10\,828\,820 \)
\( 10\,846\,286 \)
\( 10\,831\,495 \)
\( 10\,603\,879 \)

1003109204

Level: 
C
Candles are being lit at a commemoration in the square. Gradually, every \( 15 \) seconds, two candles are lit from one already burning candle. It also took \( 15 \) seconds to lit the first candle. After \( 4 \) minutes, all candles were lit. How many candles were finally burning in the square?
\( 65\,535 \)
\( 32\,767 \)
\( 32\,768 \)
\( 131\,070 \)
\( 131\,071 \)

1003109203

Level: 
C
The half-life of Francium is \( 21 \) minutes. How long will it take before its weight decreases from \( 512\,\mathrm{g} \) to \( 1\,\mathrm{g} \)?
\( 3 \) hours \( 9 \) minutes
\( 2 \) hours \( 48 \) minutes
\( 3 \) hours \( 30 \) minutes
\( 2 \) hours \( 27 \) minutes
\( 3 \) hours \( 51 \) minutes

1003109202

Level: 
C
A blouse is on sale and its price was reduced twice, each time by ten percent. The difference between its original and its final price is \( 133\,\mathrm{CZK} \). What was the blouse original price before the discounts?
\( 700\,\mathrm{CZK} \)
\( 665\,\mathrm{CZK} \)
\( 1\,330\,\mathrm{CZK} \)
\( 1\,400\,\mathrm{CZK} \)
\( 750\,\mathrm{CZK} \)