Prove that amongst any 7 different numbers it is always possible to choose two of them, \(x\) and \(y\), so that the following inequality was true: \[0 < \frac{x-y}{1+xy} < \frac{1}{\sqrt3}.\]
Find the representation of \((a+b)^n\) as the sum of \(X_{n,k}a^kb^{n-k}\) for general \(n\). Here by \(X_{n,k}\) we denote coefficients that depend only on \(k\) and \(n\).
The positive real numbers \(a, b, c, x, y\) satisfy the following system of equations: \[\left\{ \begin{aligned} x^2 + xy + y^2 = a^2\\ y^2 + yz + z^2 = b^2\\ x^2 + xz + z^2 = c^2 \end{aligned} \right.\]
Find the value of \(xy + yz + xz\) in terms of \(a, b,\) and \(c.\)
Find all solutions of the system of equations: \[\left\{ \begin{aligned} (x+y)^3=z\\ (x+z)^3=y\\ (y+z)^3=x \end{aligned} \right.\]
Find a representation as a product of \(a^{2n+1} + b^{2n+1}\) for general \(a,b,n\).
Find a representation as a product of \(a^n - b^n\) for general \(a,b,n\).
The number \(n\) is natural. Show that: \[\frac1{\sqrt{1}} +\frac1{\sqrt{2}}+ \frac1{\sqrt{3}} + \dots +\frac1{\sqrt{n}} < 3 \sqrt{n+1} -3.\]
If \(n\) is a positive integer, we denote by \(s(n)\) the sum of the divisors of \(n\). For example, the divisors of \(n=6\) are \(1,2,3,6\), so \(s(6)=1+2+3+6=12\). Prove that, for all \(n\geq1\), \[s(1)+s(2)+\cdots+s(n)\leq n^2.\] Denote by \(t(n)\) is instead the sum of the squares of the divisors of \(n\) (e.g., \(t(6)=1^2+2^2+3^2+6^2=50\)), can you find a similar inequality for \(t(n)\)?
The first player is thinking about a finite sequence of numbers \(a_1,a_2, ..., a_n\). The second player can try to find the sequence by naming his own sequence \(b_1, b_2, ...b_n\), after that the first player will tell the result \(a_1b_1 + a_2b_2 + ...a_nb_n\). In the next step the second player can say another sequence \(c_1, c_2, ...c_n\) to get another answer \(a_1c_1+ a_2c_2 + ... a_nc_n\). Find the smallest amount of steps the second player has to take to find out the sequence \(a_1,a_2,...a_n\).
I have three positive integers. When you add them together, you get \(15\). When you multiply the three numbers together, you get \(120\).
What are the three numbers?