For a natural number \(n\), we call the number \(1+2+3+\cdots + n\) the \(n^{\text{th}}\) triangular number, and we denote it by \(T_n\). Find \(T_n+T_{n-1}\) in terms of \(n\).
On a TV screen, each minute that passes, there is a number that flashes on the screen: first \(5\), then \(55\), then \(555\), and so on. Will any of these numbers be ever divisible by \(495\)? If so, which is the smallest?
For a natural number \(n\), define the \(n^{\text{th}}\) triangular number by \(T_n=1+2+3+\cdots+n\).
Show that \(3T_n+T_{n-1}\) is itself a triangular number, and determine which one.
Among \(12\) identical-looking balls, exactly one has a different weight (we do not know whether it is heavier or lighter than the others).
Using a balance scale, show how to determine the odd ball, and whether it is lighter or heavier, using only three weighings.
You are given \(68\) coins, and all of them have different weights. Using at most \(100\) weighings on a balance scale, find both the heaviest and the lightest coin.
For a complex number \(z=a+ib\), we write \(\bar z\) for its complex conjugate, defined as \(\bar z=a-ib\). Describe the set of complex numbers \(z\) such that \(z\bar z=1\).
Let \(z_1=3+5i\). Draw on the complex plane the point \(z_2=3z_1, z_3=\frac{1}2{z_1}\), and \(z_4=-z_1\). What do you notice?
Show that if \(x\) and \(y\) are complex numbers, then
\[\overline{\left(\frac{x}{y}\right)}=\frac{\bar x}{\bar y}\]
Describe the set of complex numbers \(z\) for which \(z=\bar z\).
Let points \(A,B,C,D\) on the plane be represented by complex numbers \(a,b,c,d\). Show that the segments \(AB\) and \(CD\) are parallel if and only if \[\frac{a-b}{\bar a-\bar b}=\frac{c-d}{\bar c - \bar d}.\]