9000149704
Level:
A
Find the center of the following ellipse.
\[
9x^{2} + 4y^{2} + 54x - 32y + 109 = 0
\]
\([-3;4]\)
\([-3;-4]\)
\([3;4]\)
\([3;-4]\)
A
9000149705
Level:
A
Find the center of the following ellipse.
\[
16x^{2} + 9y^{2} - 32x - 54y - 47 = 0
\]
\([1;3]\)
\([1;-3]\)
\([-1;3]\)
\([-1;-3]\)
9000148910
Level:
A
The player throws a dice in a desktop game. If he gets the number
\(6\), he
throws again and his score is the sum of both numbers. Find how many possibilities
are there for the player's score.
\(5 + 6=11\)
\(2\cdot 6=12\)
\(1 + 6=7\)
\(6\cdot 6=36\)
9000148901
Level:
A
The current Czech vehicle registration plate number has the form NLN-NNNN, where N stands
for a digit from \(0\)
to \(9\)
and L stands for a letter from an alphabet containing
\(26\)
letters. How many different registration plates are possible?
\(26\cdot 10^{6}\)
\(10^{6}\)
\(15\cdot 10^{6} + 6\cdot 10^{5}= 156\cdot 10^{5}\)
\(16\cdot 10^{6}\)
9000148903
Level:
A
A combination lock will open if a right choice of three numbers (from
\(1\) to
\(9\)) is selected.
Suppose that we use a brute force attack to open the lock (we try all possibilities). To try one
code takes \(20\)
seconds. What is the maximal time (in seconds) required to open the lock by brute
force?
\(20\cdot 9^{3}\, \mathrm{s}=14\:580\,\mathrm{s}\)
\(20\cdot \frac{9!}
{6!}\, \mathrm{s}=10\:080\,\mathrm{s}\)
\(20\cdot \frac{9!}
{3!\; 6!}\, \mathrm{s}=1\:680\,\mathrm{s}\)
\(20\cdot 9\cdot 3\, \mathrm{s}=540\,\mathrm{s}\)
9000148905
Level:
A
The king has eight daughters. A dragon asks two daughters, otherwise he will
destroy the whole country. In how many ways can we choose a pair of king's
daughters for the dragon?
\(\frac{8!}
{2!\; 6!}=28\)
\(8\cdot 7=56\)
\(8^{2}=64\)
\(8 + 7=15\)
9000148907
Level:
A
A bowl contains \(12\) different
gummy-bears and \(20\)
different sweet-drops. Anne can choose either one sweet-drop or one gummy-bear.
From the rest, Jane can choose one sweet-drop and two gummy-bears. Anne wants to
provide a maximum of the possibilities for Jane's choice. What should Anne choose?
sweet-drop
gummy-bear
Both possibilities give the same result.
9000148908
Level:
A
There are seven different yellow apples, eight different green apples and ten different
red apples. How many ways are there to choose three apples, if we wish to have three
apples of different colors?
\(10\cdot 8\cdot 7=560\)
\(\frac{10\cdot 8\cdot 7}
{2}=280 \)
\((10 + 8 + 7)\cdot 2=50\)
\(10 + 8 + 7=25\)
9000146705
Level:
A
Expand the following expression.
\[
(3 - 2a)^{2} - (3a - 4)(3a + 4)
\]
\(- 5a^{2} - 12a + 25\)
\(- 5a^{2} - 12a - 7\)
\(- 5a^{2} + 25\)
\(- 5a^{2} - 7\)
9000148906
Level:
A
Each candidate in a tender is fluent in at least one out
of two required languages (English and French). There are
\(20\) candidates fluent
in English and \(14\)
candidates fluent in French. From these amounts
\(10\)
candidates are fluent in both languages. How many candidates are there in the
tender?
\(24\)
\(34\)
\(14\)
\(44\)