please indicate how the area of the product:

(multiply 2 integer with the same sign)

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Given 2n points in a plane with no three of them collinear, Show that they can be divided into n pairs such that the n segments joining each pair do not intersect

Show that any continuous curve of unit length can be covered by a closed rectangles of area 1/4:

 A calculator is broken so that the only keys that still work are the sin, cos, tan, sin^-1, cos^-1 , and tan^-1 buttons. The display initially shows 0: Given any positive rational number q; show that pressing some finite sequence of buttons will yield q: Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

14 points are tabulated on a rectangle. Every 2 points has a distance of 1 cm. How many triangles of area 6 cm² can be formed using any of these 3 points as vertices 

A function f is difined for non - negative integers n and k as follows

f (0,n) = n+1 , f (k,0) = f (k-1,1) , f (k+1,n+1) = f (k,f (k+1,n))

Evaluate f (2,2)

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Show that if a convex quadrilateral with side-lengths a, b, c, d and area \(\sqrt{abcd}\)  has an inscribed circle, then it is a cyclic quadrilateral.

 Point P is inside \(\Delta\)ABC. Determine points D on side AB and E on side AC such that BD = CE and PD+ PE is minimum.

24 numbers can be formed using 1,2,3 and 4, where each digit is used once only in each number. Find the sum of such numbers that are divisible by 22.

BRIANA HAS3 POTS OF FLOUERS,RED,PINK AND YELLOW.SHE WANTS TO DISPLAY THEM IN THE WINDOUSILL.HOW MANY DIFF ERENT WAYS CAN SHE ARRANGE THEM?

Two circles, one of radius 8 and one of radius 18, are tangent (i.e., they intersect at exactly one point). There are two lines each of which is tangent to both circles, as shown in the diagram. What is the distance from the intersection of these lines to the center of the circle with radius 8 ?

Let S be the set of positive integers that divide at least one of the numbers 1; 11; 111; 1111; ... For example, 3 is in S since 3 divides 111 . The number of elements in S that are less than 100 is

(a) 22     (b) 28       (c) 34        (d) 40         (e) 48

Professor Derring's desk has 4 history books, 5 classical fictions and 3 lingguistic books. In how many ways can be arrange the books, if books of the same topic must be placed together?

Numbers are to be built using only digits 1,2,3,4 and 5 in such a way that each digit is only used once in each number. How many of these numbers will have the following property: The first digit is divisible by one, the first 2 digits make a number which is dividible by 2, the first 3 digits make a number which is devisible by three, the first 4 digits make a number which is divisible by 4 and all 5 digits make a number which is divisible by 5.

Given a parallelogram with the length of two sides are 9 cm and 13 cm. If the length of one height is 15 cm then the length of  the second height is ... cm.

There is a unique positive integer that has less than 11 decimal digits, ends in a 6, and if this 6 is removed and put at the front of the number (e.g., 136 → 613), then the resulting number is exactly four times the original number. How many digits does this number have? 

The figure below depicts 5 provinces of a country. In how many ways can you colour the provinces, using 4 colours, so that no neighbouring countries have the colours?

Given that 1.00000035811231² = 1.000000xyz2247482444265735361 where x, y, and z denote missing digits, what is the value of x + y + z?

Every day at noon, a ship leaves from Le Havre in France towards New York, and at the same time a ship leaves New York bound to Le Havre. The crossing takes 7 days and 7 nights. If a ship leaves Le Havre today at noon, how many ships coming from New York will it meet? 

Mrs forget the 4-digit combination to the lokk of her safe. She only remembers that none of the digits is 0 and that the sum of all the 4 digits í 9. How many times does she have to attempt to open the safe ?

We are given line segments of lengths 1, 2, 3, . . . , 99. If we have to use all the segments, is it possible to construct a square? How about a rectangle? Justify your answers.

3 identical black beads and 5 identical white beads are placed in a row. In how many ways can they be arranged so that no neighbouring beads are of black colour? 

Certain 4-digit numbers, like 1122, 3344, 3334, 6555,... are made up of 2 different digits only. How many such 4-digit numbers are there if one is to select from 1, 2, 3, 4 and 5. 

The numbers 1,2,3,4,...,10 are to be written around a circle in some order.Then each number will be added to its immediate neighbours to obtain 10 new numbers.Prove that 16 is the largest possible value of the smallest of these new numbers.

A regular 13-sided polygon is inscribed in a circle with center O.Triangles can be formed by choosing 3 vertices of this polygon to be the vertices of the triangle.Prove that there are 91 triangles formed in this way with the point O inside the triangle.

Yurko saw a tractor slowly pulling a long pipe down the road. Yurko walked along beside the pipe in the same direction as the tractor,and counted 140 paces to get from one end to the other.Then he turned around and walked back to the other end,taking only 20 paces.The tractor and Yurko kept to a uniform speed and Yurko's paces were all 1 m long.Prove that the pipe is 35 m long

Several non-overlapping isosceles triangles have vertex O in common.Every triangle shares an edge with each immediate neighbour.The smallest of the angles at O has size m⁰ (\(m\in Z^+\)) and the other triangles have angles at O of size 2m⁰,3m⁰,4m⁰ and so on.The diagram shows an arrangement of 5 such triangles.Prove that 3 is the smallest value of m for such a set of triangles exists.

Gene and Anne have 1 coin and continue to flip it until one of them wins the game.Gene wins if the sequence "Head-Head" appears before the sequence "Tail-Head". Anne wins if the sequence "Tail-Head" appears before the sequence "Head-Head".What is Anne's probability of winning this game ?

A.1/4        B.1/3            C.1/2           D.2/3             E.3/4

The positive integers a,b and c satisfy \(\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{c^2}\)

The sum of all possible values of \(a\le100\) is :

A.315          B.615                C.680           D.555          E.620

The number 12 has six factors: 1,2,3,4,6 and 12. Four of these are even (2,4,6 and 12) and two are odd (1 and 3). 

(1) Find some numbers which have all their factors, except 1, even. Describe the sequence of numbers that has this property.

(2) Find some numbers which have exactly half their factors even. Again describe the sequence of numbers that has this property. 

The base of a triangular piece of paper ABC is 12 cm long.The paper is folded down over the base,with the crease DE parallel to the base of the paper.The area of the triangle that projects below the base is 16% that of the area of the triangle ABC.The length of DE,in cm,is :

A. 9.6        B. 8.4            C.7.2            D. 4.8              E.6.96

The four sides of a garden are known to be 20, 16, 12, and 10 rods, and it has the greatest possible area for those sides. What is the area? 

Here we have a perfect magic square composed of the numbers I to 16 inclusive. The rows, columns, and two long diagonals all add up 34. Now, supposing you were forbidden to use the two numbers 2 and 15, but allowed, in their place, to repeat any two numbers already used, how would you construct your square so that rows, columns, and diagonals should still add up 34? Your success will depend on which two numbers you select as substitutes for the 2 and 15. 

In the quadrilateral below, \(\angle A=60^o,\angle B=\angle D=90^o,BC=4cm\) and \(CD=6cm\). What is the length of AB?
                                        

In the pictured isosceles triangle ABC  \(\angle A=20^o\), side BE = AE and \(\angle ACD=50^o\). Without a calculator or computer find the measure of \(\angle BED\).