Odd points even triangles

This is a problem from the British (or maybe Balkan, no idea) math olympiad.


You are given a set $S$ of $2005$ points (no three collinear) in the 2D plane. For each point $P$ in $S$, you count the number of triangles (formed by points in $S$) within which $P$ lies ($P$ must be strictly inside the triangle).

Show that this number is even, irrespective of the point $P$.


[I don't remember the solution to this one, so medium.]

Sequence of product of sums and reciprocal sums

Similar to the last two problems, but no triangles this time.

Suppose $t_1, t_2, \dots, t_n, \dots$ are a sequence of positive reals.

Let

$$S_n = \sum_{i=1}^{n} t_i = t_1 + t_2 + \dots + t_n$$

and

$$R_n = \sum_{i=1}^{n} \frac{1}{t_i} = \frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n}$$


Show that:

If $$S_k R_k \lt k^2 + 1$$ for infinitely many $k$, then all the $t_i$s must be equal.

More about sides of a triangle

This problem is inspired by an IMO (don't know which year) and was the inspiration for the previous problem.

Anyway. Here goes:

Suppose $t_1, t_2, \dots, t_n$ are $n \ge 3$ positive real numbers such that

$$ (t_1 + t_2 + \dots + t_n)\left(\frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n}\right) \lt n^2 + 1$$

Show that:

Problem 1) For any distinct $i, j, k$ the numbers $t_i, t_j, t_k$ form the sides of some triangle. [This is the original IMO problem]

Problem 2) If $n \ge 13$, show that the triangles are all acute.

Problem 3) If $n \le 12$, the triangles are not necessarily acute.

A property of sides of a triangle?

Prove or disprove:

Proposition: $a, b, c$ are sides of a triangle if

$$ (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \lt 10$$

Note that we can easily show that (using AM $\ge$ GM for instance)

$$ (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \ge 9$$

for any positive reals $a, b, c$.

So we can restate the above statement as

Proposition: $a, b, c$ are sides of a triangle if

$$\left\lfloor (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \right\rfloor = 9$$

where $\lfloor x \rfloor$ is the integer part of $x$.


[Note: Initially I had if only if, but I have changed the statements to just be an implication]

An integral with $\frac{1}{\log x}$

Suppose $n$ is a positive integer (though the result below does not really need that).

Show that

$$ \int_{0}^{1} \frac{x^n - 1}{\log x} \text{d}x = \log(n+1)$$

Note that the $\log x$ is the $\log$ to base $e$.

Ant in a room

This is a classic puzzle.

There is an ant in one of the corners of a cubic room (say of side 10ft). It wants to get to the diagonally opposite corner.

What is the shortest distance it can travel to achieve that? Note that it can only walk on the wall/floor (it cannot fly etc).

Partscores are important too...

This is a hand from one the sectional swiss teams in the Seattle area.

You are south and end up in 2S (bidding below).

IMPS N/S	North ♠ J2 ♥ T432 ♦ T432 ♣ J32
	South ♠ KQ9843 ♥ AQ ♦ A5 ♣ 654

W	N	E	S
		1C	1S
1NT	P	P	2S
P	P	P

Opponents cash 3 clubs, and RHO then switches to a low trump at trick 4. What will you do?

This is probably a simple hand as a puzzle.


You should stick in the 9 in case LHO holds AT7x.

If LHO has AT7x, then to defeat this contract, LHO needs to duck and not cover with the T. Maybe LHO will talk themselves into covering by thinking of some trump promotion scenario.

At the table LHO covered, and now it was easy to win the J, take the heart finesse and draw trumps, making 2 (losing 3 clubs, 1 diamond and 1 spade) [LHO did have AT7x].

If LHO does not cover, you could hope trumps are 3-2, and overtake the 9 with the J to take the heart finesse. [Or you could win the 9 in hand and play spade to J, hoping LHO will win and return a heart].

At the other table, the defence to first 4 tricks was the same (3 clubs, spade switch by East), but declarer didn't put in the 9 (or 8) and went down. 5 IMPs.