The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.
A group of psychologists developed a test, after which each person gets a mark, the number \(Q\), which is the index of his or her mental abilities (the greater \(Q\), the greater the ability). For the country’s rating, the arithmetic mean of the \(Q\) values of all of the inhabitants of this country is taken.
a) A group of citizens of country \(A\) emigrated to country \(B\). Show that both countries could grow in rating.
b) After that, a group of citizens from country \(B\) (including former ex-migrants from \(A\)) emigrated to country \(A\). Is it possible that the ratings of both countries have grown again?
c) A group of citizens from country \(A\) emigrated to country \(B\), and group of citizens from country \(B\) emigrated to country \(C\). As a result, each country’s ratings was higher than the original ones. After that, the direction of migration flows changed to the opposite direction – part of the residents of \(C\) moved to \(B\), and part of the residents of \(B\) migrated to \(A\). It turned out that as a result, the ratings of all three countries increased again (compared to those that were after the first move, but before the second). (This is, in any case, what the news agencies of these countries say). Can this be so (if so, how, if not, why)?
(It is assumed that during the considered time, the number of citizens \(Q\) did not change, no one died and no one was born).
In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.
All the positive fractions smaller than \(1\) with denominators not more than \(100\) are written in a row. Isley and Ella put signs \("+"\) or \("-"\) in front of any fraction, which does not yet have a sign before it. They write signs in turns, but it is known that Isley has to make the last move and calculate the resulting sum. If the total sum turns out to be an integer number, then Ella will give her a chocolate bar. Will Isley be able to get a chocolate bar regardless of Ella’s actions?
Cambria was building various cuboids from \(1\times 1\times1\) cubes. She initially built one cuboid, then increased its length and width by \(1\) and reduced its height by \(2\). She then understood that she needs the same number of \(1\times 1\times 1\) cubes to build both the original and new cuboids. Prove that the number of cubes used for each of the cuboids is divisible by \(3\).
Using areas of squares and rectangles, show that for any positive real numbers \(a\) and \(b\), \((a+b)^2 = a^2+2ab+b^2\).
The identity above is true for any real numbers, not necessarily positive, in fact in order to prove it the usual way one only needs to remember that multiplication is commutative and the distributive property of addition and multiplication:
\(a\times b = b\times a\);
\((a+b)\times c = a\times c + b\times c\).
Annie found a prime number \(p\) to which you can add \(4\) to make it a perfect square. What is the value of \(p\)?
Let \(a\) and \(b\) be positive real numbers. Using areas of rectangles and squares, show that \(a^2 - b^2 = (a-b) \times (a+b)\).
Try to prove it in two ways, one geometric and one algebraic.
Let \(a\) and \(b\) be positive real numbers. Using volumes of cubes and parallelepipeds, show that \((a+b)^3 = a^3 +3a^2b+3ab^2 +b^3\).
Hint: Place the cubes with sides \(a\) and \(b\) along the same diagonal.