C

1003158506

Level: 
C
Let there be the first nine terms of an arithmetic sequence with the first term \( a_1=1 \) and the difference \( d=1 \). Suppose we are creating ordered triples of \( 3 \) different numbers out of the given \( 9 \) numbers so that they form \( 3 \) consecutive terms of a geometric sequence. How many of such triples can be created?
\( 8 \)
\( 6 \)
\( 4 \)
\( 3 \)
\( 9 \)

1003158505

Level: 
C
Let there be three numbers that are \( 3 \) consecutive arithmetic sequence terms. Their sum is \( 9 \). If the first number is divided by \( -3 \), we get \( 3 \) consecutive terms of a geometric sequence. Find the largest number of the given three.
\( 9 \)
\( 3 \)
\( 12 \)
\( 6 \)
\( 4 \)

1003158504

Level: 
C
Let there be three numbers that are three consecutive arithmetic sequence terms with a difference of \( d=3 \). If the third number is decreased by \( \frac32 \), we get \( 3 \) consecutive terms of a geometric sequence. Find the third number (of an arithmetic sequence).
\( 0 \)
\( 3 \)
\( -3 \)
\( \frac32 \)
\( -\frac32 \)

1003158503

Level: 
C
Let there be four numbers, such that the first three numbers form consecutive arithmetic sequence terms with a difference of \( d=-6 \), and the last three numbers form consecutive terms of a geometric sequence with the common ratio \( q=\frac23 \). Find the fourth number.
\( 8 \)
\( 18 \)
\( 12 \)
\( -24 \)
\( -4 \)

1003158502

Level: 
C
Let there be two unknown positive numbers between numbers \( 12 \) and \( 54 \). The first three numbers of these four form \( 3 \) consecutive arithmetic sequence terms, and the last \( 3 \) numbers form three consecutive terms of a geometric sequence. Find the smaller of the two unknown numbers.
\( 24 \)
\( 36 \)
\( 15 \)
\( 20 \)
\( 32 \)

1003158501

Level: 
C
Let \( 3 \) numbers be \( 3 \) consecutive terms of a geometric sequence with the common ratio \( q=4 \). If we increase the second number by \( 9 \), we get \( 3 \) consecutive arithmetic sequence terms. Find the first number.
\( 2 \)
\( 4 \)
\( 8 \)
\( 16 \)
\( 32 \)

1103170608

Level: 
C
Let there be an equilateral triangle with a side length of \( 16\,\mathrm{cm} \). Connecting the centres of its sides with line segments we form another equilateral triangle. If we add two more triangles using the same procedure, what is the sum of their areas?
\( 85\sqrt3\,\mathrm{cm}^2 \)
\( 128\sqrt3\,\mathrm{cm}^2 \)
\( \frac{341}4\sqrt3\,\mathrm{cm}^2 \)
\( 90\,\mathrm{cm}^2 \)
\( 148\sqrt3\,\mathrm{cm}^2 \)

1103170607

Level: 
C
Let there be an equilateral triangle with a side length of \( 16\,\mathrm{cm} \). Connecting the centres of its sides with line segments we form another equilateral triangle. If we add three more triangles using the same procedure, what is the sum of their perimeters?
\( 93\,\mathrm{cm} \)
\( 72\,\mathrm{cm} \)
\( 144\,\mathrm{cm} \)
\( 31\,\mathrm{cm} \)
\( 90\,\mathrm{cm} \)

1003170605

Level: 
C
Suppose we insert number \( 10\,530 \) between \( 2 \) unknown numbers such that they form \( 3 \) consecutive terms of a geometric sequence and their sum is \( 31\,707 \). What is the smaller of the two unknown numbers?
\( 9\,477 \)
\( 10\,500 \)
\( 9\,832 \)
\( 10\,034 \)
\( 5\,265 \)
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