The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).
The teacher wrote on the board in alphabetical order all possible \(2^n\) words consisting of \(n\) letters A or B. Then he replaced each word with a product of \(n\) factors, correcting each letter A by \(x\), and each letter B by \((1 - x)\), and added several of the first of these polynomials in \(x\). Prove that the resulting polynomial is either a constant or increasing function in \(x\) on the interval \([0, 1]\).
The graph of the function \(y=kx+b\) is shown on the diagram below. Compare \(|k|\) and \(|b|\).
We are given a polynomial \(P(x)\) and numbers \(a_1\), \(a_2\), \(a_3\), \(b_1\), \(b_2\), \(b_3\) such that \(a_1a_2a_3 \ne 0\). It turned out that \(P (a_1x + b_1) + P (a_2x + b_2) = P (a_3x + b_3)\) for any real \(x\). Prove that \(P (x)\) has at least one real root.
There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:
Alex is 1 year older than V,
Beatrice is 2 years older than W,
Victor is 3 years older than X,
Gregory is 4 years older than Y.
Who is older and by how much: Deborah or Z?
Compare the numbers: \(A=2011\times 20122012\times 201320132013\) and \(B= 2013\times 20112011 \times 201220122012\).
We are given \(n+1\) different natural numbers, which are less than \(2n\) (\(n>1\)). Prove that among them there will always be three numbers, where the sum of two of them is equal to the third.
Let \(x_1, x_2, \dots , x_n\) be some numbers belonging to the interval \([0, 1]\). Prove that on this segment there is a number \(x\) such that \[\frac{1}{n} (|x - x_1| + |x - x_2| + \dots + |x - x_n|) = 1/2.\]
51 points were thrown into a square of side 1 m. Prove that it is possible to cover some set of 3 points with a square of side 20 cm.