Two players are playing a game. The first player is thinking of a finite sequence of positive integers \(a_1\), \(a_2\), ..., \(a_n\). The second player can try to find the first player’s sequence by naming their own sequence \(b_1\), \(b_2\), ..., \(b_n\). After this, the first player will give the result \(a_1b_1 + a_2b_2 + ...+a_nb_n\). Then the second player can say another sequence \(c_1\), \(c_2\), ..., \(c_n\) to get another answer \(a_1c_1+ a_2c_2 + ... +a_nc_n\) from the first player. Find the smallest number of sequences the second player has to name to find out the sequence \(a_1\), \(a_2\), ..., \(a_n\).
In the sum below, different letters denote different digits and the same letters denote the same digit. \[P.Z + T.C + D.R + O.B + E.Y\] None of the five terms are integers, but the sum itself is an integer. Find the possible sums of the expression. For each possible answer, write one example with these five terms. Explain why other sums cannot be obtained.
Show that there are no rational numbers \(a,b\) such that \(a^2 + b^2 = 3\).
Suppose \(x,y\) are real numbers such that \(x < y + \varepsilon\) for every \(\varepsilon > 0\). Show that \(x \leq y\).
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?
Can you find a formula relating \(1^3+2^3+\dots+n^3\) to \(1+2+\dots+n\)?
Prove the reverse triangle inequality: for every pair of real numbers \(x\), \(y\), we have \(\left| \left| x \right| - \left| y \right| \right| \leq \left| x - y \right|\).
For every natural number \(k\ge2\), find two combinations of \(k\) real numbers such that their sum is twice their product.