Problems
Three complex number \(a,b,c\) are called affinely independent if whenever \(t,s,u\) are real numbers such that \(ta+sb+uc = 0\) and \(t+s+u=0\), we have that \(t=s=u=0\). Show that three complex numbers \(a,b,c\) are affinely independent if and only if they are not collinear.
Let \(\triangle ABC\) be a triangle and \(A'\) be the midpoint of the side \(BC\). The segment \(AA'\) is a called a median of \(\triangle ABC\). Similarly, there are two more medians constructed from \(B\) and \(C\). Show that the three medians intersect at a point and give a formula for that point in terms of the three vertices. This point is called the centroid of \(\triangle ABC\).
The three altitudes of a triangle intersect at a point called the orthocenter of the triangle. Suppose that the vertices of a \(\triangle ABC\) lie on a circle of radius 1 centered at 0. Show that the centroid, the orthocenter and the circumcenter of \(\triangle ABC\) are collinear. This line is called the Euler line of the triangle. Note that the circumcenter of \(\triangle ABC\) is just 0 by our assumption.
Now the detective has actually \(126\) people involved in the case! But he is in a big rush and needs to solve the case in only \(9\) days. Can you help him?
Which number is larger \(1234567\times 1234569\) or \(1234568^2\)?
Which of the two fractions is larger? \[\frac{1\overbrace{00\cdots 00}^{1984\text{ zeroes}}1 }{1\underbrace{00\cdots 000}_{1985\text{ zeroes}}1}\qquad \text{or}\qquad \frac{1\overbrace{00\cdots 00}^{1985\text{ zeroes}}1 }{1\underbrace{00\cdots 000}_{1986\text{ zeroes}}1}\]
Which is larger? \[95^2+96^2\qquad \text{or}\qquad 2\times 95\times 96\]
Among all rectangles with perimeter \(4\), show that the one with largest area is a square, and determine that largest area.
Show how the following diagram “proves" that \((a+b)^2=a^2+2ab+b^2\) without just expanding the brackets:
