9000153802
Level:
A
Find the number of mutually different isosceles triangles (at least
two sides are equal) such that each side of each triangle is either
\(2\),
\(3\),
\(4\) or
\(5\).
\(14\)
\(16\)
\(17\)
\(24\)
Combinatorics
9000153803
Level:
A
Find the number of mutually different triangles such that all
three sides of each triangle are mutually different and each side is
\(2\),
\(3\),
\(4\) or
\(5\).
\(3\)
\(4\)
\(14\)
\(16\)
9000153806
Level:
A
Determine the number of three-digit positive integers that can be formed using the digits \(2\),
\(3\),
\(4\) and
\(5\). The digits can be used repeatedly.
\(64\)
\(24\)
\(256\)
\(81\)
9000153808
Level:
A
Find the number of subsets of the set
\(\{2,3,4,5\}\).
\(16\)
\(4\)
\(6\)
\(15\)
9000153807
Level:
A
Find the number of the positive integers with three digits which can be written using
the digits \(2\),
\(3\),
\(4\) and
\(5\). Each
digit can be used at most once.
\(24\)
\(64\)
\(256\)
\(81\)
9000148904
Level:
A
Pamela needs new ski for a ski course. There are skis from six different vendors in a
shop. The shop has four different ski pairs from each vendor, but two vendors have all
products behind Pam's financial limit. How many pairs are at disposal for Pam?
\(4\cdot 4=16\)
\(4!=24\)
\(4\cdot 2=8\)
\(4 + 2=6\)
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?
\(\frac{8!}
{2!\; 6!}=28\)
\(8\cdot 7=56\)
\(8^{2}=64\)
\(8 + 7=15\)