Problems

You start with the number \(1\) on a piece of paper. You may perform two operations:

double up (multiply the number on your paper by \(2\), and erase the old number), increase by \(1\) (add \(1\) to the number on your paper, and erase the old number).

So for example, you may end up with the numbers \(1,2,3,6,\cdots\). Show that it is possible to obtain \(975\) starting from \(1\) in at most \(16\) operations.

A collection of weights is made from the weights \(1,2,4,8,\dots\) grams (that is, all powers of \(2\)). Some weights may appear several times. The weights are placed on the two pans of a balance scale and the scale is in balance. It is known that all the weights on the left pan are different.

Prove that the number of weights on the right pan is at least as large as the number of weights on the left pan.

\(CD\) is a chord of a circle with centre \(A\). The line \(CD\) is parallel to the tangent to the circle at the point \(B\). Prove that the triangle \(BCD\) is isosceles.

Four lines, intersecting at the point \(D\), are tangent to two circles with a common center \(A\) at the points \(C,F\) and \(B,E\). Prove that there exists a circle passing through all the points \(A,B,C,D,E,F\).

A circle with center \(A\) is inscribed into the triangle \(CDE\), so that all the sides of the triangle are tangent to the circle. We know the lengths of the segments \(ED=c, CD=a, EC=b\). The line \(CD\) is tangent to the circle at the point \(B\) - find the lengths of segments \(BD\) and \(BC\).

A circle with center \(A\) is tangent to all the sides of the quadrilateral \(FGHI\) at the points \(B,C,D,E\). Prove that \(FG+HI = GH+FI\).

Let \(a\) be a positive number. Which is larger: \(a^2\) or \(a\)? And which is larger: \(a^2 + 1\) or \(a\)?

The numbers \(a\), \(b\) and \(c\) are positive. By completing the square, show that \[\frac{a^2}4 + b^2 + c^2 \ge ab-ac+2bc.\]

Let \(m\) and \(n\) be natural numbers such that \(m>n\). Show that: \[\frac1{n^2} + \frac1{(n+1)^2} + \frac1{(n+2)^2} + \dots + \frac1{m^2} > \frac1{n} - \frac1{m}.\]

The numbers \(a,b,c\) are positive. Show that: \[\frac{ab}{c} + \frac{bc}{a} + \frac{ac}{b} \ge a +b+c.\]