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#### TrueTears

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##### Fun questions :)
« on: April 23, 2009, 05:52:59 pm »
+1
1.The remainder when $1^3 + 2^3 + 3^3 + 4^3 + ... + 100^3$ divided by 7 is?

2. Three-digit numbers are formed using only digits which are odd. The sum of all such three-digit numbers is?

3. Find the limit as x approaches infinity of $\frac{x}{2^x}$ without using l'hopital's rule.
(source: kamil9876)

4. The Fibonacci ratios are defined as follows: $\frac{1}{1},\frac{2}{1},\frac{3}{2},\frac{5}{3}$ (i.e $r_n=\frac{f_{n+1}}{f_{n}}$ )
prove that $r_1r_3 etc. (increase and decrease alternatively)
(source: kamil9876)

5. Consider $f(n)=i^0 + i^{\frac{4}{k}} + i^{\frac{2*4}{k}} + i^{\frac{3*4}{k}}...+i^{\frac{n*4}{k}}$
What shape is made when f(0) is connected to f(1), f(1) to f(2) etc.
(source: kamil9876)

6. A time machine is controlled by a bank of switches, numbered 1 to 10 (from left to right). Each switch must be set to 0 or 1 before the starter button is pressed. The nth switch takes the time traveller forward $2^{n-1}$ years if n is odd and back $2^{n-1}$ years if n is even; a switch set to 0 has no effect; the combined effect of several switches set to 1 is the sum of their individual effects. How should the switches be set to take the time traveller back to 1787, the year in which the First Fleet departed? (assuming we start in year 1988)

7. The number of digits in $200^{1988}$ when expanded is closest to?
A. 450 B. 4000 C. 4500 D. 6500 E. 40000

8. Ten circles, each circle intersecting each other in two distinct points, are in a plane and no three of them pass through the same point. They divide the plane into a number of disjoint regions which, including the exterior of all circles, is?

9. In how many ways can one select 3 different integers ranging from 1 to 30 inclusive so that their sum is a multiple of 3 (order of selection does not matter)?

10. Find all the real roots to $x^4 - 40x^3 + 206x^2 -40x +1 = 0$

11. A point is selected inside a rectangle such that its distance from one vertex is 11cm, its distance from the opposite vertex is 12cm, and its distance from a third vertex is 3cm. Its distance, in cm, from the fourth vertex is?

12. In how many ways can 75 be expressed as the sum of at least two positive integers, all of which are consecutive?

13. How many times in one day do the hands of a clock form a right angle? (Assume the hour hand does not move, eg, say on a real clock when it is 5:50 pm, the hour hand tends to move towards 6. But assume for this question the hour hand stays exactly on the hour it is.)

14. In a circle of radius 10 cm, a chord is drawn 6 cm from the centre. If a chord half the length of the original chord were drawn, its distance, in cm, from the centre would be?

15. In the triangle PQR, T is a point on PQ such that PT is twice as long as TQ and U is a point on QR such that QU is twice as long as UR. If the area of triangle PQR is $90cm^2$, then the area of triangle PTU in $cm^2$ is?

16. Let $a$ and $b$ be positive real numbers with $a>b$. Prove that

$\lim_{n \to \infty} ^n\sqrt{{a^n+b^n}}=a$

17. Show that for all natural numbers n, every number of the form$2^{n+1} + 3^{2n-1}$ is divisible by $7$.
(source: kamil9876)

18. A slow train leaves Canberra at 9:17 am and arrives in Goulburn at 12:02pm. On the same day, the express leaves Canberra at 9:56am and arrives in Goulburn at 11:36am. At what time does the express pass the slow train if each is travelling at constant speed?

19. An honest promoter decides to stage an exhibition tennis match between 2 players, whose names are to be drawn at random from a pool of N players, n of whom are Australians. He draws the 2 names and before announcing them is asked by a reporter "Will the match involve at least one Australian?" After looking at the drawn names he says "Yes". The chance of both players being Australian is?

20. A cube of edge 6cm is divide into 216 unit cubes by planes parallel to the faces of the cube. A sphere of diameter 6 cm is inscribed in the large cube so that faces of this cube are tangents to the sphere. The number of complete unit cubes contained in this sphere is?

21. Dots are arranged in a rectangular grid containing 4 rows and n columns. Consider different ways of colouring the dots, colouring each dot either yellow or green. Call a colouring of the diagram "good" if no four dots of the same colour form a rectangle (or square) with horizontal and vertical sides. The maximum value of n for which the diagram permits a "good" colouring is?

22. Mao, over9000, /0, TrueTears and kamil play a game in which each person is a frog or ugly duckling. Frogs' statements are always false while ugly ducklings' statements are always true.
Mao says that over9000 is a ugly duckling.
/0 says that TrueTears is a frog
kamil says that Mao is not a frog
over9000 says that /0 is not a ugly duckling
TrueTears says that kamil and Mao are different kinds of animals
How many frogs are there?

23. Each of the faces of a cube is numbered with a different integer. each vertex is assigned a "vertex number" which is the sum of the numbers on the faces which intersect in that vertex and then the sum of the vertex numbers is calculated. The highest number which must divide this sum, for every possible numbering of the faces, is?

24. 4 spheres each of radius 10 cm lie on a horizontal table so that the centre of the spheres form a square of side 20cm. A fifth sphere of radius 10 cm is placed on them so that it touches each of the spheres without disturbing them. How many centimetres above the table is the centre of the 5th sphere?

25. In a rectangular co-ordinate system with scale 1 cm = 1 unit on each axis, we have A(0,8) B(6,0) C(x,12). If $0 and area of triangle ABC is $20cm^2$. Find the value of x
(source: Chinese textbook lol)

26. On a tiny, remote island where the death sentence still exists, a man can be granted mercy after receiving a death sentence in the following way: He is given 18 white balls and 6 black balls. He must divide them among three boxes, with at least one ball in each box. The blindfolded, he must choose a box at random, and then a ball from within this box. He receives mercy only if the chosen ball is white. The probability of the man receiving mercy (provided he has distributed the balls in the most favourable manner) is?

27. PQR is a triangle. Q is the mid-point of PS. The length of UR is $\frac{2}{3}$ the length of PU. T is the point of intersection of the lines QR and SU. The ratio $\frac{QT}{QR}$ is?

28. Find $\sqrt{-\sqrt{3}+\sqrt{4+\sqrt{5}+\sqrt{17-4\sqrt{15}}}}$
(source: /0)

29. A city railway network has for sale one-way tickets for travel from one station to another station. Each ticket specifies the origin and destination. Several new stations were added to the network and an additional 76 different ticket types had to be printed. How many new stations were added to the network?

30. How many real solutions are there to $\frac{x^3}{\sqrt{4-x^2}} + x^2 - 4 = 0$

31. Given integers m and n such that $1 \le m < n$, how many positive solutions must the equation $-1 - x^m + x^n = 0$ have?

32. Two brothers, each aged between 10 and 90, "combined" their ages by writing them down one after the other to create a four digit number and discovered this number to be the square of an integer. Nine years later they repeated the process (combining their names in the same order) and found that the combination again was a square of another integer which was nine larger than the previous integer. The sum of their original ages were?

33. When a school marching band lines up in 4 columns, the last row is not complete. When it lines up in 3 columns, the last row is not complete, but there are 3 more rows. When it lines up in 2 columns, the last row is not complete and there are 5 more rows than there were with 3 columns. The number of students in the band is between?
A. 10-19 B. 20-29 C. 30-39 D. 40-49 E. 50-59

34. For how many positive integers n is it true that $\frac{n+17}{n-7}$ is also a positive integer?

35. The numbers p, q, r, s and t are consecutive positive integers, arranged in increasing order. If p+q+r+s+t is a perfect cube and q+r+s is a perfect square, find the smallest possible value of r.

36. Find all integers that leave a remainder of 3  when divided by 5 , a remainder of 5  when divided by 7 , and a remainder of 7  when divided by 11.
(sauce: /0)

37. If the line y = 2x+3 is reflected in the line y = x+1, find what its equation becomes.
(sauce: /0)

38. What is the smallest integer n for which n! terminates in exactly 88 zeros?
(sauce: /0)

39. I start with 3 marbles, one red, one green and one blue. I can trade any one marble for two others, one each of the other two colours. Is it possible to make a number of such trades and end up with five more blue marbles than red? Show all working. (I don't care how many green marbles I have at the end.)

40. Take any whole number q. Calculate $q^2-1$. Factorize $q^2-1$ to give two factors a and b (not necessarily $q+1$ and $q-1$). Put $c=a+b+2q$. Then you will find that $ab+1$, $bc+1$ and $ca+1$ are all perfect squares. Prove that this method always gives three perfect squares.

41. Semicircles are drawn on the sides of a rectangle ABCD as shown in the diagram. A circle passing through points ABCD carves out four crescent-shaped regions (coloured yellow and green in the diagram). Prove that the sum of the areas of the four crescents is equal in area to the rectangle ABCD.

42. Prove $x^n+y^n = z^n$ has no solution for $n=3$, with $x,y,z$ integers.

43. Consider a rectangle $l$ units long, and $w$ units wide and the $lw$ unit squares found in it. A diagonal is drawn from one corner to the opposite. The diagonal cuts through certain unit squares. Find in terms of $l$ and $w$ the number of unit squares that are cut through.
(sauce: kamil9876)

44. If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

45. A job needs three men to work for two weeks (10 working days).
Andy works for all 10 days.
Bert works for the first week and Clive works for the second week.
Dave works for 6 days, but then is too sick to work.
Eddy takes his place for 3 days, then Fred does the last day.
When the job is finished they are all paid the same amount. At first they could not work out how much each man should have, but then Fred says: "If I give my wages to Andy, and Eddy gives 100 to both Dave and me, then the wages will be correct". How much was paid for the whole job, and how much does each man get?

46.  The green triangle is an isosceles triangle while the blue triangle is an equilateral triangle. Find a in terms of b and c. What can you say about the triangles if a=b=c?

47. Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal (45 degree cuts) through any point inside the square?

48. Three people (Alan, Ben and Chris), collectively own a certain number of gold sovereigns. Respectively they own a half, one third and one sixth of the total number. All the sovereigns were piled on a table and each of them grabbed a part of the pile so that none were left.
After a short time:
Alan returned half of the sovereigns that he had taken.
Ben returned a third of what he had taken.
Chris returned one sixth of what she had taken.
Finally each of the three got an equal amount of what had been returned to the table. Surprisingly, each person had exactly the number of sovereigns that really belonged to them. What is the smallest number of sovereigns that this strange transaction will work for? How much did each person grab from the pile?

49. The diagram shows a rectangular box (a cuboid). The areas of the faces are 3, 12 and 25 square centimetres. What is the volume of the box? Or more generally, the areas of the faces of a cuboid are p , q  and r . What is the volume of the cuboid in terms of p , q and r?

50. Two motorboats are travelling up and down a lake at constant speeds and turning at each end without slowing down. They leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A and, for the second time, 400 metres from B. How long is the lake (the distance from A to B) and what is the ratio of the speeds of the boats?

END OF FIRST 50 QUESTIONS.
« Last Edit: May 01, 2009, 05:22:21 pm by TrueTears »
PhD @ MIT (Economics).

Interested in asset pricing, econometrics, and social choice theory.

#### Over9000

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##### Re: Fun questions :)
« Reply #1 on: April 23, 2009, 06:34:48 pm »
0
2. 111,113,115,117,119,131,133,135,137,139,151.......
311,313,315,317,319,331,333,335,337,339,351.......
511.....
711....
911............................999
So from this we can see, there will be $(5 \times 5 \times 5)= 125$ of these certain 3 digit odd numbers. Obtained from the logic that for example, in the 100's for starters, for each group of 10 (i.e 111...119), there are 5 of these digits, and in every 100 (i.e 111.....199) there are 5 groups of suitable 10's (i.e 111..,131....,151....,171.....191....), and from 111 to 999, there are 5 groups of 100's that are possible (i.e 111...,311....,511.... and so on).
so using arithmetic series formula
$Sn= \frac{n}{2}(a + l)$
$Where a=111, l=999$ and $n=125$
$Sn= \frac{125}{2}(111 + 999)$
$Sn= \frac{125}{2}(1110)$
$Sn= 125 \times 555$
using multiplication on paper which I cbf typing
$Sn= 69375$
« Last Edit: April 25, 2009, 08:02:38 pm by Over9000 »
Gundam 00 is SOOOOOOOOHHHHHHHHH GOOOOOOOOOOODDDDDDDDDDDD I cleaned my room

VCE 200n(where n is an element of y): Banter 3/4, Swagger 3/4, Fresh 3/4, Fly 3/4

#### TrueTears

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##### Re: Fun questions :)
« Reply #2 on: April 23, 2009, 06:36:54 pm »
0
51. Prove that the area of the annulus is $\frac{1}{4}\pi(AB)^2$ where AB is a tangent to the inner circle.

52. Consider the equation $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ where a, b and c are natural numbers and $0. Prove that $a<3$ and also that $b<4$ and hence that there is only one set of values which satisfy this equation.

An extension to this problem: Find the six sets of values which satisfy the equation $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1$ where a, b, c and d are natural numbers and $0.

53. Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
Extension: Show that if a convex polygon has more than six sides, then at least one of the sides has an obtuse angle at both ends.

54. A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base. Likewise show that 10101 is composite in any base.

55. Let $f_0(x) = \frac{1}{1-x}$ and $f_n(x) = f_0(f_{n-1}(x))$ where n = 1, 2, 3, 4... Evaluate $f_{2000}(2000)$

56. Is it true that any convex hexagon will tessellate if
$\implies$ It has a pair of opposite sides that are equal, and
$\implies$ Three adjacent angles that add up to 360 degrees?
You may assume that the adjacent angles that add up to 360 degrees are "in between" the equal sides, as shown in the diagram.

Can you show that it will always be true? What about a counter example?

57. Find the missing angle

58. Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

59. ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

60. The shape shown is of an equilateral triangle sitting on top of a square. What is the radius of the circle that circumscribes this shape?

61. The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

62. A small circle is placed inside a square just touching the sides. The square is on top of a big circle with its vertices on the circle. Finally the big circle is on top of a trapezium which has its sides just touching the circle. Using the measurements and clue given, find the area of the trapezium.

63. Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.

64. A one hectare field is in the shape of a right angled triangle. At the midpoint of each side is a post. over9000 the goat is tethered at the midpoint of the hypotenuse, and two sheep, TT and Mao, at the midpoints of the other two sides. Each rope is just exactly long enough to allow the animal to reach two adjacent vertices. One and only one animal can reach all points in the field. Which one is it and why?

65. Find the maximum of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$ where $p$, $q$ and $r$ are positive integers and $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$. Prove that it is indeed a maximum.

66. Show that if you add 1 to the product of four consecutive numbers the answer is always a perfect square.

67. What is the remainder when $2^{164}$ is divided by $7$?

68. Simplify $(1+\tan25^{\circ})(1+\tan20^{\circ})$
(sauce: /0)

69. Find $\frac{b}{a}$ if $\log_{9}a=\log_{15}b=\log_{25}(a+2b)$
(sauce: /0)

70. If $a+b+c=6$, then prove that $\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2 \geq \frac{75}{4}$
(sauce: /0)

71. Solve the equation $1!+2!+3!+...+x!=y^2$ for integer pairs $(x,y)$.
(sauce: /0)

72. Evaluate $\sum_{n=0}^\infty \frac{\cos{(n\theta)}}{2^n}$ for $cos(\theta)=\frac{1}{5}$
(sauce: /0)

73. The graph of $x^2+y^2 =1$ is rotated about the line $y=-1$. Find the surface area of the solid formed.
(sauce: /0)

74. You got 3 red marbles and N blue marbles, how many ways are there of arranging all the marbles in a straight row such that red marbles don't touch.
(sauce: kamil)
« Last Edit: June 20, 2009, 03:22:26 pm by TrueTears »
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#### /0

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##### Re: Fun questions :)
« Reply #3 on: April 23, 2009, 07:55:46 pm »
0
The formula for summing cubes is

$\sum_{i=1}^n = \left(\frac{n(n+1)}{2}\right)^2$

So we have the division as $\frac{\left(\frac{100(100+1)}{2}\right)^2}{7} = \frac{25502500}{7}$

The remainder through division is 2

To find the remainder without doing that, refer to:
http://www.artofproblemsolving.com/Wiki/index.php/Modular_arithmetic/Introduction
We have:
$1^3 \equiv 1 \pmod 7$,
$2^3 \equiv 1 \pmod 7$,
$3^3 \equiv -1\pmod 7$,
$4^3\equiv (-3)^3 \equiv 1 \pmod 7$,
$5^3\equiv (-2)^3 \equiv -1 \pmod 7$,
$6^3 \equiv (-1)^3=-1 \pmod 7$,
$7^3 \equiv 0^3 = 0 \pmod 7$,
So we know that $1^3+...+7^3 \equiv 1+1-1+1-1-1 \equiv 0 \pmod 7$
So $1^3+...+98^3 \equiv 14(1^3+...+7^3) \equiv 0 \pmod 7$

$99^3+100^3 \equiv 1^3+2^3 \equiv 1+8 \equiv 9 \equiv 2 \pmod 7$ so the remainder is 2.

#### TrueTears

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##### Re: Fun questions :)
« Reply #4 on: April 23, 2009, 07:59:44 pm »
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Very nice /0
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#### TrueTears

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##### Re: Fun questions :)
« Reply #5 on: April 23, 2009, 08:41:18 pm »
0
3. $2^x = e^{xln2}$

$\lim_{x \to \infty}(\frac{x}{2^x}) = \lim_{x \to \infty}(x \times e^{-xln2}) = 0$
« Last Edit: April 23, 2009, 08:42:52 pm by TrueTears »
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#### kamil9876

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##### Re: Fun questions :)
« Reply #6 on: April 23, 2009, 09:12:51 pm »
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3. $2^x = e^{xln2}$

$\lim_{x \to \infty}(\frac{x}{2^x}) = \lim_{x \to \infty}(x \times e^{-xln2}) = 0$
right answer, though there is a gap in ur logic. the exponential term tends to 0, however the x term tends to infinity so u cant treat this as a product of two limits. (aka u dont know if the exponential completely dominates the x such that the expression can become smaller than any $\epsilon>0$)
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

#### TrueTears

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##### Re: Fun questions :)
« Reply #7 on: April 23, 2009, 09:24:17 pm »
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$f(x)=\frac{x}{2^x}$
Consider the pattern formed by $f(1),f(2),f(3)$ etc..

The numerator and denominators are changing. the denominator is being multiplied by 2 each time you go to the next one. However when you go from f(x) to f(x+1) the numerator is being multiplied by: $\frac{x+1}{x}=1+\frac{1}{x}$. But this decreases as x approaches infinity:

The limit can be written as:

$\frac{1}{2}*\frac{\frac{2}{1}}{2}*\frac{\frac{3}{2}}{2}*\frac{\frac{4}{3}}{2}...$
$=\frac{\frac{3}{2}}{2}*\frac{\frac{4}{3}}{2}...$
$<\frac{\frac{3}{2}}{2}*\frac{\frac{3}{2}}{2}*\frac{\frac{3}{2}}{2}....$(because the numerators in this product are bigger)
$=0$
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#### Mao

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##### Re: Fun questions :)
« Reply #8 on: April 23, 2009, 10:13:58 pm »
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3.$x = 2^{\log_2 (x)}$

then, $\frac{x}{2^x} = 2^{\log_2 (x) - x}$

as $x\to \infty, \; x >> \log_2 (x)$ (rate of increase of log approaches zero as x increases)

$\implies \lim_{x\to \infty} 2^{\log_2 (x) - x} = \lim_{t\to \infty} 2^{-t} = 0$

$\implies \lim_{x\to \infty} \frac{x}{2^x} = 0$
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#### TrueTears

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« Reply #9 on: April 25, 2009, 11:28:03 am »
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New questions posted !
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#### TrueTears

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##### Re: Fun questions :)
« Reply #10 on: April 25, 2009, 11:34:43 am »
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Alternate solution to q 1.

Any number $x^3$ can be written as $(a+7n)^3$, where n is an integer and $a \in [1,2, ...,7]$

Expanding $(a+7n)^3 = a^3 + 3a^2(7n) +3a(7n)^2 +(7n)^3$

So remainder when divided by 7 is the same remainder when diving $a^3$ by 7

Now {$1^3, 2^3 ....7^3$} have remainders {$1,1,6,1,6,6,0$}. These remainders add to 21 which is equivalent to a remainder of 0.

So $(1^3+ ...7^3), (8^3 + ...14^3), (15^3 + ... +21^3)...$ all have remainders of 0

$\therefore 1^3 +2^3 + ...+ 100^3 = (1^3 + ...+7^3) + (8^3 + ... + 14^3) + ... +99^3 + 100^3 = 0 + 0 + ... + 1 + 1 = 2$
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#### Mao

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##### Re: Fun questions :)
« Reply #11 on: April 25, 2009, 11:39:52 am »
0
$\int_{-\mbox{pr0}}^{\mbox{pr0}} \mbox{pr0}\; d(\mbox{pr0}) = \left[ \frac{\mbox{pr0}^2}{2}\right]_{-\mbox{pr0}}^{\mbox{pr0}} = 0$....
« Last Edit: April 25, 2009, 05:49:14 pm by Mao »
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#### TrueTears

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« Reply #12 on: April 25, 2009, 11:43:40 am »
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$\int_{-\mbox{pr0}}^{\mbox{pr0}} \mbox{pr0}\; d(\mbox{pr0}) = \left[ \frac{\mbox{pr0}^2}{2}\right]_{-\mbox{pr0}}^{\mbox{pr0}} = \mbox{pr0}^2$....
roflmao $lim_{\mbox{mao} \to \mbox{sunnies}} f(\mbox{overcoat}) = \mbox{NEO}$
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#### enwiabe

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##### Re: Fun questions :)
« Reply #13 on: April 25, 2009, 12:03:10 pm »
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$\int_{-\mbox{pr0}}^{\mbox{pr0}} \mbox{pr0}\; d(\mbox{pr0}) = \left[ \frac{\mbox{pr0}^2}{2}\right]_{-\mbox{pr0}}^{\mbox{pr0}} = \mbox{pr0}^2$....

Actually you get ((pr0)^2)/2 - ((pr0)^2)/2 = 0 ...

#### kamil9876

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##### Re: Fun questions :)
« Reply #14 on: April 25, 2009, 01:32:33 pm »
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9.)
The three numbers can be split into 3 groups of 10. :

set of numbers a such that$a (mod 3)=0$
set of numbers b such that$b (mod 3)=1$
set of numbers c such that$c (mod 3)=2$

There are 4 different ways for the sum of three numbers to equal 0 modulo 3:

case1:0+ 0+ 0=0=0 mod 3
case2:2+ 2+ 2=6=0 mod 3
case3:1+ 1+ 1=3=0 mod 3
case4:0 +1 +2=3=0 mod 3

number of different ways for case1: 10C3
Same for case 2 and 3.
For case 1 the number of different ways is 10*10*10:

Hence the total number of ways is:

$3*(10C3)+10^3=\frac{3*9*8*7}{1*2*3}+10^3$

Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."