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
\(5\).
\(14\)
\(16\)
\(17\)
\(24\)
A
9000375402
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation does not have a solution.
\[
2x + a = a(a^{2} - x)
\]
\(\left \{-2\right \}\)
\(\left \{1\right \}\)
\(\left \{-1\right \}\)
\(\left \{0\right \}\)
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
\(5\).
\(3\)
\(4\)
\(14\)
\(16\)
9000375403
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has infinitely many solutions.
\[
a^{2}x + ax - a = 2x - 1
\]
\(\left \{1\right \}\)
\(\emptyset \)
\(\left \{0\right \}\)
\(\left \{-2\right \}\)
9000168701
Level:
A
Which of the given points is one of the vertices on the major axis of the ellipse \(9x^{2} + 16y^{2} - 18x + 96y + 9 = 0\)?
\([5;-3]\)
\([4;-3]\)
\([1;1]\)
\([1;0]\)
9000375404
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has infinitely many solutions.
\[
a^{2}x + 2ax - 3x = a - 2
\]
\(\emptyset \)
\(\left \{-3;1\right \}\)
\(\left \{-3;1;2\right \}\)
\(\left \{0\right \}\)
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.
\(64\)
\(24\)
\(256\)
\(81\)
9000375405
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has a unique solution.
\[
a^{2}x + 6x = a + 1 - 5ax
\]
\(\mathbb{R}\setminus \left \{-3;-2\right \}\)
\(\mathbb{R}\setminus \left \{2;3\right \}\)
\(\mathbb{R}\setminus \left \{-1;2;3\right \}\)
\(\mathbb{R}\setminus \left \{-3;-2;1\right \}\)
9000154808
Level:
A
Little John plays a dice game against Robin Hood. To win, he needs to get the sum of \(8\) by rolling two dice. What is the probability that he wins over Robin right on the first roll? Round your result to three decimal places.
\(0{.}139\)
\(0{.}194\)
\(0{.}806\)
\(0{.}778\)
9000153808
Level:
A
Find the number of subsets of the set
\(\{2,3,4,5\}\).
\(16\)
\(4\)
\(6\)
\(15\)