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As shown in the figure, circle of diameter 4 touches the other two circles and the bottom of the rectangle.

Circle of diameter 3 touches the top at A and left side of the rectangle.

Circle of diameter 6 touches the top at B and two other sides of the rectangle.

Find the length of AB.

\(AB = x+y\)

\(\begin{array}{|rcll|} \hline (1.5+2)^2 &=& x^2+(6-1.5-2)^2 \\ 3.5^2 &=& x^2+2.5^2 \\ x^2 &=& 3.5^2-2.5^2 \\ x^2 &=& 6 \\ x &=& \sqrt{6} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline (2+3)^2 &=& y^2 + (3-2)^2 \\ 5^2 &=& y^2 +1 \\ y^2 &=& 25-1\\ y^2 &=& 24 \\ y^2 &=& 4*6 \\ y &=& 2\sqrt{6} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline AB &=& x+y \\ AB &=& \sqrt{6}+2\sqrt{6} \\ \mathbf{AB} &=& \mathbf{3\sqrt{6}} \\ \hline \end{array}\)

Guest I think you may have misread the question. 

(unless it was me of course)  

Find the number of 3-digit numbers, where any two adjacent digits differ by 1.

Let the first digit by 2,3,4,5,6, or 7

Then there will be 6*2*2=24 such numbers

Say the first digit is 1 or 8 

101

121

123

898

878

876

there are 6 of those

Say the first digit is 9

987

989

there are 2 of those

So that is   24+6+2=32

Find the sum 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + ... + 301 + 302.  
(The pattern of signs is + + - - + + - - ...)

\(\begin{array}{|rll|} \hline && 1 \\ && \underbrace{+2 - 3}_{-1} \underbrace{ - 4 + 5}_{+1}\\ && \underbrace{ + 6 - 7}_{-1} \underbrace{ - 8 + 9}_{+1} \\ && + \ldots \\ && \underbrace{+298 - 299}_{-1} \underbrace{-300 + 301}_{+1} \\ && + 302 \\ &=& 1+0+302 \\ &=& \mathbf{303} \\ \hline \end{array} \)

non zero, 2 of the next 3digits are 0

There are 9999-1000+1 = 9000 numbers altogether.

9 of them have 3 zeros

How many have 2 zeros

there are 9 choices for the fist digit, there are 9 choices for the other non-zero digit, there are 3 places there it can go

so

9*9*3 = 243 

So I think the answer is  9+243 = 252

Guests gets 171, maybe I have double counted or maybe guest missed some.

Guest, have you included 1001 ??

There are 13 such numbers:

100001  100010  100100  101000  110000  200000  1000001  1000010  1000100  1001000  1010000  1100000  2000000  Total = 13 such numbers

For a certain arithmetic progression, the sum of the first 1729 terms is equal to the sum of the first 29 terms. 

Find, with proof, the sum of the first 1758 terms. 

Show all your steps!

Formula arithmetic progeression: \(a_n = a_1 + (n-1)d\)

\(\begin{array}{|rcll|} \hline s_{1729} &=& \left(\dfrac{a_1+a_{1729}}{2}\right)*1729 \quad & | \quad a_{1729} = a_1 + 1728d \\ s_{1729} &=& \left(\dfrac{a_1+a_1 + 1728d}{2}\right)*1729 \\ s_{1729} &=& \left(\dfrac{2a_1+ 1728d}{2}\right)*1729 \quad & | \quad \text{the sum of the first 1729 terms} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline s_{29} &=& \left(\dfrac{a_1+a_{29}}{2}\right)*29 \quad & | \quad a_{29} = a_1 + 28d \\ s_{29} &=& \left(\dfrac{a_1+a_1 + 28d}{2}\right)*29 \\ s_{29} &=& \left(\dfrac{2a_1+ 28d}{2}\right)*29 \quad & | \quad \text{the sum of the first 29 terms} \\ \hline \end{array} \)

\(\mathbf{s_{1729}=s_{29}}\)

\(\begin{array}{|rcll|} \hline \mathbf{s_{1729} } &=& \mathbf{s_{29}} \\\\ \left(\dfrac{2a_1+ 1728d}{2}\right)*1729 &=& \left(\dfrac{2a_1+ 28d}{2}\right)*29 \\ \left( 2a_1+ 1728d \right)*1729 &=& \left( 2a_1+ 28d \right)*29 \\ 2*1729a_1+1728*1729d &=& 2*29a_1+28*29d \\ 2a_1(1729-29) &=& (28*29-1728*1729)d \\ 2a_1*1700 &=& -2986900d \\ 2a_1*17 &=& -29869d \\ \mathbf{ 2a_1 } &=& \mathbf{-1757d} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline s_{1758} &=& \left(\dfrac{a_1+a_{1758}}{2}\right)*1758 \quad & | \quad a_{1758} = a_1 + 1757 \\ s_{1758} &=& \left(\dfrac{a_1+a_1 + 1757d}{2}\right)*1758 \\ s_{1758} &=& \left(\dfrac{2a_1+ 1757d}{2}\right)*1758 \quad & | \quad \text{the sum of the first 1758 terms} \\ && \boxed{\mathbf{ 2a_1 = -1757d}} \\ s_{1758} &=& \left(\dfrac{-1757d+ 1757d}{2}\right)*1758 \\ s_{1758} &=& \left(\dfrac{0}{2}\right)*1758 \\ \mathbf{s_{1758}} &=& \mathbf{0} \\ \hline \end{array}\)

The sum of the first 1758 terms is 0.

There are 171 such numbers:

1000  1010  1020  1030  1040  1050  1060  1070  1080  1090  1100  1200  1300  1400  1500  1600  1700  1800  1900  2000  2010  2020  2030  2040  2050  2060  2070  2080  2090  2100  2200  2300  2400  2500  2600  2700  2800  2900  3000  3010  3020  3030  3040  3050  3060  3070  3080  3090  3100  3200  3300  3400  3500  3600  3700  3800  3900  4000  4010  4020  4030  4040  4050  4060  4070  4080  4090  4100  4200  4300  4400  4500  4600  4700  4800  4900  5000  5010  5020  5030  5040  5050  5060  5070  5080  5090  5100  5200  5300  5400  5500  5600  5700  5800  5900  6000  6010  6020  6030  6040  6050  6060  6070  6080  6090  6100  6200  6300  6400  6500  6600  6700  6800  6900  7000  7010  7020  7030  7040  7050  7060  7070  7080  7090  7100  7200  7300  7400  7500  7600  7700  7800  7900  8000  8010  8020  8030  8040  8050  8060  8070  8080  8090  8100  8200  8300  8400  8500  8600  8700  8800  8900  9000  9010  9020  9030  9040  9050  9060  9070  9080  9090  9100  9200  9300  9400  9500  9600  9700  9800  9900  Total =  171

There are 214 such numbers:

100  101  102  103  104  105  106  107  108  109  110  112  120  121  122  123  124  125  126  127  128  129  132  134  143  145  154  156  165  167  201  210  211  212  213  214  215  216  217  218  219  221  223  230  231  232  233  234  235  236  237  238  239  243  245  254  256  265  267  301  310  312  320  321  322  323  324  325  326  327  328  329  332  334  340  341  342  343  344  345  346  347  348  349  354  356  365  367  401  410  412  421  423  430  431  432  433  434  435  436  437  438  439  443  445  450  451  452  453  454  455  456  457  458  459  465  467  501  510  512  521  523  532  534  540  541  542  543  544  545  546  547  548  549  554  556  560  561  562  563  564  565  566  567  568  569  601  610  612  621  623  632  634  643  645  650  651  652  653  654  655  656  657  658  659  665  667  701  710  712  721  723  732  734  743  745  754  756  760  761  762  763  764  765  766  767  768  769  801  810  812  821  823  832  834  843  845  854  856  865  867  901  910  912  921  923  932  934  943  945  954  956  965  967  Total =  214

3736 mod 24 = 16 hours

16 + 2 AM =18 - 12 = 6 PM  3736 hours from now.

101  113  131  137  139  151  157 = 7 such numbers  

The sides of a triangle are 37, 684, and 685.  Find its area.

\(\begin{array}{|rcll|} \hline \sqrt{37^2+684^2} &=& 685 \\ \hline \end{array}\)

This is a right angular triangle!

The area is: \( \dfrac{37*684}{2} = 12654 \)

Compute \(2*\arctan\left(\dfrac{1}{3}\right) + \arctan\left(\dfrac{1}{7}\right)\).

\(\begin{array}{|rcll|} \hline && 2*\arctan\left(\dfrac{1}{3}\right) + \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left(\dfrac{1}{3}\right)+\arctan\left(\dfrac{1}{3}\right) + \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left( \dfrac{1}{3} + \dfrac{1}{3} \above 1pt 1-\dfrac{1}{3}\cdot \dfrac{1}{3} \right)+ \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left( \dfrac{3}{4} \right)+ \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left( \dfrac{3}{4} + \dfrac{1}{7} \above 1pt 1-\dfrac{3}{4}\cdot \dfrac{1}{7} \right)+ \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan(1) \\ &=& 45^\circ \quad & | \quad \left(\dfrac{\pi}{4}\right) \\ \hline \end{array}\)

If x^2 + 1/x^2 = 171, then what are the possible values of x - 1/x?

\(\begin{array}{|rcll|} \hline \left( x - \dfrac{1}{x} \right)^2 &=& x^2-2x\dfrac{1}{x} + \dfrac{1}{x^2} \\ \left( x - \dfrac{1}{x} \right)^2 &=& x^2+ \dfrac{1}{x^2} -2 \quad &|\quad x^2 + \dfrac{1}{x^2} = 171 \\ \left( x - \dfrac{1}{x} \right)^2 &=&171 -2 \\ \left( x - \dfrac{1}{x} \right)^2 &=&169 \\ x - \dfrac{1}{x} &=& \pm \sqrt{169} \\ \mathbf{ x - \dfrac{1}{x} } &=& \mathbf{\pm 13} \\ \hline \end{array} \)

All A are B.

One-third of all D are A.

Half of all C are B.

One C is a A.

Eight C are D.

There are 90 B.

Draw the Venn diagram. How many B are neither A nor C?

I am not convinced that there is enough into to sove this.

Here is a Venn diagram outline.  This is how I see the basic problem.  

I will continue to think about it.

3) A/B = 10/9.

4) BZ = 15/2.

The coordinates of A are (sqrt(2),1/2).