Classic trig equality

Show that

$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$



[Solution (math.stackexchange link).]

Yet another inequality

$x_1, x_2, \dots, x_n$ are $n$ positive real numbers with sum $S$, $n \gt 1$.

Show that

$$\sum_{i=1}^{n} \frac{x_i}{S - x_i} \ge \frac{n}{n-1}$$


Solution [Click here to expand/collapse]

Using AM >= GM we have that

$\sum y_i \sum \frac{1}{y_i} \ge n^2$

Applying this to (wlog, assuming $S = 1$) $$\left(\sum_{i=1}^{n} \frac{1}{1-x_i}\right) \left(\sum_{i=1}^{n} (1 - x_i)\right) \ge n^2$$ i.e $$\sum_{i=1}^{n} \frac{1}{1-x_i} \ge \frac{n^2}{n-1}$$ And so $$\sum_{i=1}^{n} \left(\frac{1}{1-x_i} - 1\right) \ge \frac{n^2}{n-1} -n$$ Which gives us what we want to prove.

Splitting naturals into arithmetic progressions

The set of natural numbers $\{1,2,\dots\}$ is split into finite number of arithmetic progressions with common differences $d_1 \le d_2 \le \dots \le d_n$

Show that $$d_{n} = d_{n-1}$$

Eg: $\{4n+1\}, \{4n+3\}, \{2n\}$

Here $d_1 = 2, d_2 = d_3 = 4$.

Pointed nines

This is a hand from Mercercrest bridge club.

You are South. RHO opens 1D, you preempt 2S and opponents end up in 6H.

Partner leads the SK.

MPS Both
		Dummy ♠ 975 ♥ A9 ♦ AT542 ♣ KQ2
	You ♠ AQT862 ♥ 32 ♦ QJ3 ♣ 85

W	N	E	S
		1D	2S
3H	P	3S	P
4C	P	4H	P
4NT	P	5H	P
6H

3S was some kind of stopper asking bid.

Do you have a decision to make a trick one?


Looks like declarer has a singleton spade for the 4NT bid.

Declarer likely has 6+ hearts the DK and CA. If declarer has two or fewer diamonds, then declarer is cold.

So give declarer exactly 3 diamonds: a hand like ?, KQJxxx, K9x, Axx

Now if declarer has the singleton SJ, then you get squeezed in diamonds and spades, thanks to the diamond and spade nines!

So give partner the SJ.

Now if you play low, partner could potentially continue the SJ and now you are in charge of protecting spades and get squeezed again!

So you overtake with the A and declarer follows low. Now you play the SQ having partner keep the SJ.

Now when declarer draws trump throw your ST at the first opportunity! This will force partner to hold on to the SJ and the squeeze will not operate.

 

Mercercrest card reading hand

Mercercrest bridge club in Mercer island has its games every Tuesday evening and is considered one of the better fields in the Seattle area.

This is a hand from a recent game.

You are South and end up in 2S after the following auction

MPS None	Dummy ♠ xxx ♥ Tx ♦ KTx ♣ AJTxx
	You ♠ JT98x ♥ AJxx ♦ A9x ♣ x

W	N	E	S
1C	P	1H	1S
X*	2S	P	P
P

X was support, promising exactly 3 hearts. E/W play that 1D typically promises 4 and play 2/1 with strong NT (15-17).

LHO starts off with the Spade AKQ of trumps, RHO following twice and throwing a club on the third.

LHO then shifts to a low heart you play low from dummy and RHO plays low. Your spots are low enough that you have to win the J.

How will you play?



At the table I decided to play a club to the A and ruff a club back in hand. RHO followed with the Q.

The hand is almost an open book now.

LHO didn't open 1D, so cannot be 3=3=4=3, thus LHO has the CK. LHO also had the AKQ of spades. Since RHO played low on the heart, it is likely that RHO does not have both the K and Q.

LHO also has a balanced hand. If they had the HK they would have 15 points and thus would need both the DQ and DJ otherwise they would have opened 1NT. This would give RHO HQ and CQ.

So it is likely that LHO has the HQ and RHO has both the DQ and DJ (otherwise LHO would open 1NT).

Thus LHO has AKQ, Qxx, xxx, Kxxx and RHO had xx, Kxxx, QJxx, Qxx.


You can guarantee the contract by playing A and another heart.

If LHO wins and shifts to a diamond, thanks to your T and 9, you can capture RHO's Jack and then endplay them with a heart to lead a diamond back.

If RHO overtakes the HQ and cashes their 4th heart they are endplayed into leading a diamond back.

As it happened, LHO won the HQ and continued the CK!

Now you can ruff go to dummy with the DK and cash the good club to squeeze RHO for the overtrick. Made 9.

The four hands

MPS None	Dummy ♠ xxx ♥ Tx ♦ KTx ♣ AJTxx
West ♠ AKQ ♥ Qxx ♦ xxx ♣ Kxxx		East ♠ xx ♥ Kxxx ♦ QJxx ♣ Qxx
	You ♠ JT98x ♥ AJxx ♦ A9x ♣ x

W	N	E	S
1C	P	1H	1S
X*	2S	P	P
P

Extra chance?

This is a hand (with small changes) from the recent Bothell KO sectional (swiss teams).


You are South (and vul) and hold xx, KQT9xx, AKT, JT. RHO passes, 2H by partner which you pass and LHO balances with a double. Partner bids 3D (!), RHO bids 3S and you decide to bid 4H.

LHO leads a low spade and you see:

IMPS N/S	North ♠ Qx ♥ Axx ♦ 98xxxx ♣ Kx
	South ♠ xx ♥ KQT9xx ♦ AKT ♣ JT

W	N	E	S
			1H
P	2H	P	P
X	3D	3S	4H
P	P	P

RHO wins the SK and shifts to the DJ.

If you draw trumps they will be 2-2. How will you play?





Looks like the DJ is a singleton (LHO made a take out double) and RHO is angling for a ruff or two.

You have 9 tricks (assuming CA onside) and there seems to be no way to get the 10th trick, unless defense makes a mistake.

Is there a way to improve your chances of having the defense make a mistake?

Win the DJ cash HKQ drawing trumps and play the CJ!

If LHO ducks this (playing you for xx, KQxxxx, AK, JTX) then you are through. Play the CK and exit a club.

Either diamonds are 2-2, or the person who wins the last spade will be endplayed!

Escalator length

Two people go down an escalator (which is also moving down) taking one step at a time. The first person takes a step three times faster than the second. If the first person takes 75 steps total to reach the bottom and the second takes 50 steps, how many steps are showing in the escalator?

Removing perfect squares

Apparently this is from topcoder, but I believe there must be a more original source for this.

You start with $n$ cards numbered $1,2, \dots n$ placed in order along a line.

Now you make a pass through the cards and remove any that have a perfect square on them. Then you renumber the cards as $1,2 \dots K$ (making sure to maintain the ordering) and keep doing the process of removal and renumbering till there is only one card left.

What was the original number of that card? Can you give a formula in terms of $n$?