Problems

Prove that a $10\times 10$ board cannot be covered by T-shaped tiles (shown below)

Zara has an $8\times 8$ chessboard, in the usual coloring. She can repaint all the squares of a row or column, i.e., all white squares become black, and all black squares become white. Can she get exactly one black square?

A rectangular floor is to be covered by $2\times 2$ and $1\times 4$ tiles (everything is arranged). Unfortunately one tile got smashed, but we have one more tile of the other kind available. Can we retile the floor perfectly?

There are 13 green, 15 blue, and 17 red chameleons on an island. Whenever two chameleons of different colours meet, they both swap to the third colour (i.e., a green and blue would both become red). Is it possible for all chameleons to become one colour?

Numbers 1 and 2 are written on a whiteboard. Every day Louise’s friend Zara changes these numbers to their arithmetic mean $a_m$ and harmonic mean $h_m$.

(The arithmetic mean of two numbers $a$ and $b$ is $a_m=\frac{a+b}{2}$, and harmonic mean of two numbers $a$ and $b$ is $h_m=\frac{2}{\frac{1}{a}+\frac{1}{b}}$ ).

(a) At some point Zara wrote $\frac{941664}{665857}$ as one of the two numbers (it is not known which). What was the other number written on the whiteboard at that time?

(b) Can $\frac{35}{24}$ be ever written by Zara on the whiteboard?

Using mathematical induction prove that $$1+2+3+\cdots +n=\frac{n(n+1)}{2}.$$

There are $n$ lines on a plane, and all the lines intersect at exactly one point. Prove that the lines divide the plane into $2n$ parts.

There are $n$ lines on a plane, no two lines are parallel, and no three lines cross at one point. Show that those lines dived the plane into $\frac{n(n+1)}{2}+1$ regions.

In a sequence 2, 6, 12, 20, 30, ... find the number

(a) in the 6th place

(b) in the 2016th place.

Using mathematical induction prove that $$1+3+5+\cdots +(2n-1)=n^2.$$