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Question Number 16839 Answers: 0 Comments: 2

∫x^e^x dx

$$\int\mathrm{x}^{\mathrm{e}^{\mathrm{x}} } \mathrm{dx} \\ $$

Question Number 16834 Answers: 0 Comments: 3

If α<β<γ<2π and cos (x+α)+cos (x+β)+cos (x+γ)=0 for all x∈R, then is γ−α=((2π)/3)?

$$\mathrm{If}\:\alpha<\beta<\gamma<\mathrm{2}\pi\:\mathrm{and} \\ $$$$\mathrm{cos}\:\left({x}+\alpha\right)+\mathrm{cos}\:\left({x}+\beta\right)+\mathrm{cos}\:\left({x}+\gamma\right)=\mathrm{0} \\ $$$$\mathrm{for}\:\mathrm{all}\:{x}\in\mathbb{R},\:\mathrm{then}\:\mathrm{is} \\ $$$$\gamma−\alpha=\frac{\mathrm{2}\pi}{\mathrm{3}}? \\ $$

Question Number 16823 Answers: 0 Comments: 2

solve for g. 6(4^x + g^x ) = 13.6^x

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{g}. \\ $$$$\mathrm{6}\left(\mathrm{4}^{\mathrm{x}} \:+\:\mathrm{g}^{\mathrm{x}} \right)\:=\:\mathrm{13}.\mathrm{6}^{\mathrm{x}} \\ $$

Question Number 16836 Answers: 1 Comments: 0

Find how many number greater than 2,500 can be formed from the digit 0, 1, 2, 3, 4 if no digit can be used more than once.

$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:\mathrm{number}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{2},\mathrm{500}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{the}\:\mathrm{digit} \\ $$$$\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\:\mathrm{if}\:\mathrm{no}\:\mathrm{digit}\:\mathrm{can}\:\mathrm{be}\:\mathrm{used}\:\mathrm{more}\:\mathrm{than}\:\mathrm{once}. \\ $$

Question Number 16835 Answers: 1 Comments: 0

Question Number 16807 Answers: 1 Comments: 1

(a+2)sin α+(2a−1)cos α=(2a+1) then tan α=?

$$\left({a}+\mathrm{2}\right)\mathrm{sin}\:\alpha+\left(\mathrm{2}{a}−\mathrm{1}\right)\mathrm{cos}\:\alpha=\left(\mathrm{2}{a}+\mathrm{1}\right) \\ $$$$\mathrm{then}\:\mathrm{tan}\:\alpha=? \\ $$

Question Number 16803 Answers: 1 Comments: 1

Question Number 16794 Answers: 1 Comments: 0

Question Number 16789 Answers: 2 Comments: 1

Question Number 16785 Answers: 2 Comments: 2

Question Number 16771 Answers: 1 Comments: 1

Question Number 16788 Answers: 0 Comments: 1

Question Number 16756 Answers: 2 Comments: 1

Alternate vertices of a regular hexagon are joined as shown. What fraction of the total area of the hexagon is shaded? (Justify your answer.)

$$\mathrm{Alternate}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{hexagon} \\ $$$$\mathrm{are}\:\mathrm{joined}\:\mathrm{as}\:\mathrm{shown}.\:\mathrm{What}\:\mathrm{fraction}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{total}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hexagon}\:\mathrm{is} \\ $$$$\mathrm{shaded}?\:\left(\mathrm{Justify}\:\mathrm{your}\:\mathrm{answer}.\right) \\ $$

Question Number 16769 Answers: 0 Comments: 2

2n + p = w^a make a the subject of the formular.

$$\mathrm{2n}\:+\:\mathrm{p}\:=\:\mathrm{w}^{\mathrm{a}} \\ $$$$\mathrm{make}\:\mathrm{a}\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\mathrm{the}\:\mathrm{formular}. \\ $$

Question Number 17921 Answers: 0 Comments: 5

The value of the expression (3 − tan^2 1°)(3 − tan^2 2°)(3 − tan^2 3°)....(3 − tan^2 89°) is equal to

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\left(\mathrm{3}\:−\:\mathrm{tan}^{\mathrm{2}} \mathrm{1}°\right)\left(\mathrm{3}\:−\:\mathrm{tan}^{\mathrm{2}} \mathrm{2}°\right)\left(\mathrm{3}\:−\:\mathrm{tan}^{\mathrm{2}} \mathrm{3}°\right)....\left(\mathrm{3}\:−\:\mathrm{tan}^{\mathrm{2}} \mathrm{89}°\right) \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 17463 Answers: 2 Comments: 0

Evaluate ∫_0 ^1 ((x^4 (1−x)^4 )/(1+x^2 )) dx.

$${Evaluate}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\frac{{x}^{\mathrm{4}} \left(\mathrm{1}−{x}\right)^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}. \\ $$

Question Number 16740 Answers: 1 Comments: 3

The maximum value of cos^2 (cos (33π + θ)) + sin^2 (sin (45π + θ)) is (1) 1 + sin^2 1 (2) 2 (3) 1 + cos^2 1 (4) cos^2 2

$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{cos}^{\mathrm{2}} \:\left(\mathrm{cos}\:\left(\mathrm{33}\pi\:+\:\theta\right)\right)\:+\:\mathrm{sin}^{\mathrm{2}} \:\left(\mathrm{sin}\:\left(\mathrm{45}\pi\:+\:\theta\right)\right) \\ $$$$\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{1}\:+\:\mathrm{sin}^{\mathrm{2}} \mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{1}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{cos}^{\mathrm{2}} \mathrm{2} \\ $$

Question Number 16739 Answers: 0 Comments: 0

Let M be a point in the interior of the equilateral triangle ABC and let A′, B′ and C′ be its projections onto the sides BC, CA and AB, respectively. Prove that the sum of lengths of the inradii of triangles MAC′, MBA′ and MCB′ equals the sum of lengths of the inradii of trianges MAB′, MBC′ and MCA′.

$$\mathrm{Let}\:{M}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}\:{ABC}\:\mathrm{and}\:\mathrm{let}\:{A}', \\ $$$${B}'\:\mathrm{and}\:{C}'\:\mathrm{be}\:\mathrm{its}\:\mathrm{projections}\:\mathrm{onto}\:\mathrm{the} \\ $$$$\mathrm{sides}\:{BC},\:{CA}\:\mathrm{and}\:{AB},\:\mathrm{respectively}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{inradii}\:\mathrm{of}\:\mathrm{triangles}\:{MAC}',\:{MBA}'\:\mathrm{and} \\ $$$${MCB}'\:\mathrm{equals}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{inradii}\:\mathrm{of}\:\mathrm{trianges}\:{MAB}',\:{MBC}'\:\mathrm{and} \\ $$$${MCA}'. \\ $$

Question Number 16738 Answers: 0 Comments: 0

Prove that the segments joining the midpoints of the opposite sides of an equiangular hexagon are concurrent.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{segments}\:\mathrm{joining}\:\mathrm{the} \\ $$$$\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{equiangular}\:\mathrm{hexagon}\:\mathrm{are}\:\mathrm{concurrent}. \\ $$

Question Number 16737 Answers: 0 Comments: 0

A convex hexagon is given in which any two opposite sides have the following property: the distance between their midpoints is ((√3)/2) times the sum of their lengths. Prove that the hexagon is equiangular.

$$\mathrm{A}\:\mathrm{convex}\:\mathrm{hexagon}\:\mathrm{is}\:\mathrm{given}\:\mathrm{in}\:\mathrm{which} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{have}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{property}:\:\mathrm{the}\:\mathrm{distance} \\ $$$$\mathrm{between}\:\mathrm{their}\:\mathrm{midpoints}\:\mathrm{is}\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{times}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{lengths}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{hexagon}\:\mathrm{is}\:\mathrm{equiangular}. \\ $$

Question Number 16736 Answers: 0 Comments: 0

The side lengths of an equiangular octagon are rational numbers. Prove that the octagon has a symmetry center.

$$\mathrm{The}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equiangular} \\ $$$$\mathrm{octagon}\:\mathrm{are}\:\mathrm{rational}\:\mathrm{numbers}.\:\mathrm{Prove} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{octagon}\:\mathrm{has}\:\mathrm{a}\:\mathrm{symmetry} \\ $$$$\mathrm{center}. \\ $$

Question Number 16735 Answers: 0 Comments: 0

Let a_1 , a_2 , ..., a_n be the side lengths of an equiangular polygon. Prove that if a_1 ≥ a_2 ≥ ... ≥ a_n , then the polygon is regular.

$$\mathrm{Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:...,\:{a}_{{n}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{of}\: \\ $$$$\mathrm{an}\:\mathrm{equiangular}\:\mathrm{polygon}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if} \\ $$$${a}_{\mathrm{1}} \:\geqslant\:{a}_{\mathrm{2}} \:\geqslant\:...\:\geqslant\:{a}_{{n}} ,\:\mathrm{then}\:\mathrm{the}\:\mathrm{polygon}\:\mathrm{is} \\ $$$$\mathrm{regular}. \\ $$

Question Number 16734 Answers: 0 Comments: 0

An equiangular polygon with an odd number of sides is inscribed in a circle. Prove that the polygon is regular.

$$\mathrm{An}\:\mathrm{equiangular}\:\mathrm{polygon}\:\mathrm{with}\:\mathrm{an}\:\mathrm{odd} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{inscribed}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{polygon}\:\mathrm{is}\:\mathrm{regular}. \\ $$

Question Number 17477 Answers: 0 Comments: 0

Given (4xy/x^2 +y^2 )dy/dx=1, y=0, x=0 show that (√x)(x^2 −5y^2 )=1