Valentina added a number (not equal to 0) taken to the power of four and the same number to the power two and reported the result to Peter. Can Peter determine the unique number that Valentina chose?
Anna is waiting for the bus. Which event is most likely?
\(A =\{\)Anna waits for the bus for at least a minute\(\}\),
\(B = \{\)Anna waits for the bus for at least two minutes\(\}\),
\(C = \{\)Anna waits for the bus for at least five minutes\(\}\).
In the cabinet of Anchuria there are 100 ministers. Among them there are honest and dishonest ministers. It is known that out of any ten ministers, at least one minister is dishonest. What is the smallest number of dishonest ministers there could be in the cabinet?
A square is divided into triangles (see the figure). How many ways are there to paint exactly one third of the square? Small triangles cannot be painted partially.
The pupils of class 5A had a total of 2015 pencils. One of them lost a box with five pencils, and instead bought a box with 50 pencils. How many pencils do the pupils of class 5A now have?
To test a new program, a computer selects a random real number \(A\) from the interval \([1, 2]\) and makes the program solve the equation \(3x + A = 0\). Find the probability that the root of this equation is less than \(0.4\).
It is known that \(a = x+y + \sqrt{xy}\), \(b = y + z + \sqrt{yz}\), \(c = x + z + \sqrt{xz}\). where \(x > 0\), \(y > 0\), \(z > 0\). Prove that \(a + b + \sqrt{ab} > c\).
A quadrilateral is given; \(A\), \(B\), \(C\), \(D\) are the successive midpoints of its sides, \(P\) and \(Q\) are the midpoints of its diagonals. Prove that the triangle \(BCP\) is equal to the triangle \(ADQ\).
How can you arrange the numbers \(5/177\), \(51/19\) and \(95/9\) and the arithmetical operators “\(+\)”, “\(-\)”, “\(\times\)” and “\(\div\)” such that the result is equal to 2006? Note: you can use the given numbers and operators more than once.
Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).