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Question Number 16077 Answers: 0 Comments: 0

Let ABCDE be an equiangular pentagon whose side lengths are rational numbers. Prove that the pentagon is regular.

$$\mathrm{Let}\:{ABCDE}\:\mathrm{be}\:\mathrm{an}\:\mathrm{equiangular} \\ $$$$\mathrm{pentagon}\:\mathrm{whose}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{are} \\ $$$$\mathrm{rational}\:\mathrm{numbers}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{pentagon}\:\mathrm{is}\:\mathrm{regular}. \\ $$

Question Number 16081 Answers: 2 Comments: 0

A spherical balloon of 21 cm diameter is to be filled with hydrogen at NTP from a cylinder containing the gas at 20 atmosphere at 27°C. If the cylinder can hold 2.82 litres of water, calculate the number of balloons that can be filled up.

$$\mathrm{A}\:\mathrm{spherical}\:\mathrm{balloon}\:\mathrm{of}\:\mathrm{21}\:\mathrm{cm}\:\mathrm{diameter} \\ $$$$\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{filled}\:\mathrm{with}\:\mathrm{hydrogen}\:\mathrm{at}\:\mathrm{NTP} \\ $$$$\mathrm{from}\:\mathrm{a}\:\mathrm{cylinder}\:\mathrm{containing}\:\mathrm{the}\:\mathrm{gas}\:\mathrm{at} \\ $$$$\mathrm{20}\:\mathrm{atmosphere}\:\mathrm{at}\:\mathrm{27}°\mathrm{C}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{cylinder} \\ $$$$\mathrm{can}\:\mathrm{hold}\:\mathrm{2}.\mathrm{82}\:\mathrm{litres}\:\mathrm{of}\:\mathrm{water},\:\mathrm{calculate} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{balloons}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{filled}\:\mathrm{up}. \\ $$

Question Number 16075 Answers: 0 Comments: 0

Prove that the perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the opposite sides are concurrent.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{perpendiculars} \\ $$$$\mathrm{dropped}\:\mathrm{from}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral}\:\mathrm{to}\:\mathrm{the}\:\mathrm{opposite} \\ $$$$\mathrm{sides}\:\mathrm{are}\:\mathrm{concurrent}. \\ $$

Question Number 16074 Answers: 0 Comments: 0

Let K, L, M and N be the midpoints of the sides AB, BC, CD and DA, respectively, of a cyclic quadrilateral ABCD. Prove that the orthocenters of the triangles AKN, BKL, CLM and DMN are the vertices of a parallelogram.

$$\mathrm{Let}\:{K},\:{L},\:{M}\:\mathrm{and}\:{N}\:\mathrm{be}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{AB},\:{BC},\:{CD}\:\mathrm{and}\:{DA}, \\ $$$$\mathrm{respectively},\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral} \\ $$$${ABCD}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangles}\:{AKN},\:{BKL},\:{CLM}\:\mathrm{and} \\ $$$${DMN}\:\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{parallelogram}. \\ $$

Question Number 16072 Answers: 1 Comments: 3

Let ABCD be a convex quadrilateral. Prove that the orthocenters of the triangles ABC, BCD, CDA and DAB are the vertices of a quadrilateral congruent to ABCD and prove that the centroids of the same triangles are the vertices of a cyclic quadrilateral.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangles}\:{ABC},\:{BCD},\:{CDA}\:\mathrm{and}\:{DAB} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{quadrilateral} \\ $$$$\mathrm{congruent}\:\mathrm{to}\:{ABCD}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{centroids}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{triangles}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral}. \\ $$

Question Number 16071 Answers: 1 Comments: 0

Let A′, B′ and C′ be points on the sides BC, CA and AB of the triangle ABC. Prove that the circumcircles of the triangles AB′C′, BA′C′ and CA′B′ have a common point. Prove that the property holds even if the points A′, B′ and C′ are collinear.

$$\mathrm{Let}\:{A}',\:{B}'\:\mathrm{and}\:{C}'\:\mathrm{be}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{sides} \\ $$$${BC},\:{CA}\:\mathrm{and}\:{AB}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:{ABC}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{circumcircles}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangles}\:{AB}'{C}',\:{BA}'{C}'\:\mathrm{and}\:{CA}'{B}' \\ $$$$\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{point}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{property}\:\mathrm{holds}\:\mathrm{even}\:\mathrm{if}\:\mathrm{the}\:\mathrm{points}\:{A}', \\ $$$${B}'\:\mathrm{and}\:{C}'\:\mathrm{are}\:\mathrm{collinear}. \\ $$

Question Number 16070 Answers: 0 Comments: 0

Let ABCD be a convex quadrilateral. Prove that AB.CD + AD.BC = AC.BD if and only if ABCD is cyclic (Ptolemy′s theorem).

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$${AB}.{CD}\:+\:{AD}.{BC}\:=\:{AC}.{BD} \\ $$$$\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:{ABCD}\:\mathrm{is}\:\mathrm{cyclic}\:\left(\mathrm{Ptolemy}'\mathrm{s}\right. \\ $$$$\left.\mathrm{theorem}\right). \\ $$

Question Number 16069 Answers: 0 Comments: 0

In the interior of a quadrilateral ABCD, consider a variable point P. Prove that if the sum of distances from P to the sides is constant, then ABCD is a parallelogram.

$$\mathrm{In}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{a}\:\mathrm{quadrilateral} \\ $$$${ABCD},\:\mathrm{consider}\:\mathrm{a}\:\mathrm{variable}\:\mathrm{point}\:{P}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{distances}\:\mathrm{from} \\ $$$${P}\:\mathrm{to}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{then}\:{ABCD} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{parallelogram}. \\ $$

Question Number 16068 Answers: 2 Comments: 0

Let ABCD be a convex quadrilateral and let E and F be the points of intersections of the lines AB, CD and AD, BC, respectively. Prove that the midpoints of the segments AC, BD, and EF are collinear.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{let}\:\mathrm{E}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{intersections}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lines}\:{AB},\:{CD}\:\mathrm{and} \\ $$$${AD},\:{BC},\:\mathrm{respectively}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{segments}\:{AC},\:{BD}, \\ $$$$\mathrm{and}\:{EF}\:\mathrm{are}\:\mathrm{collinear}. \\ $$

Question Number 16067 Answers: 1 Comments: 8

Let d, d′ be two nonparallel lines in the plane and let k > 0. Find the locus of points, the sum of whose distances to d and d′ is equal to k.

$$\mathrm{Let}\:{d},\:{d}'\:\mathrm{be}\:\mathrm{two}\:\mathrm{nonparallel}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{plane}\:\mathrm{and}\:\mathrm{let}\:{k}\:>\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of} \\ $$$$\mathrm{points},\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{whose}\:\mathrm{distances}\:\mathrm{to} \\ $$$${d}\:\mathrm{and}\:{d}'\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:{k}. \\ $$

Question Number 16066 Answers: 2 Comments: 0

Let ABCD be a convex quadrilateral and let k > 0 be a real number. Find the locus of points M in its interior such that [MAB] + 2[MCD] = k.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{let}\:{k}\:>\:\mathrm{0}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{points}\:{M}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\left[{MAB}\right]\:+\:\mathrm{2}\left[{MCD}\right]\:=\:{k}. \\ $$

Question Number 16065 Answers: 0 Comments: 0

Let ABCD be a convex quadrilateral. Find the locus of points M in its interior such that [MAB] = 2[MCD].

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{points}\:{M}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left[{MAB}\right]\:=\:\mathrm{2}\left[{MCD}\right]. \\ $$

Question Number 16064 Answers: 0 Comments: 8

Let ABCD be a convex quadrilateral and M a point in its interior such that [MAB] = [MBC] = [MCD] = [MDA]. Prove that one of the diagonals of ABCD passes through the midpoint of the other diagonal.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{M}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left[{MAB}\right]\:=\:\left[{MBC}\right]\:=\:\left[{MCD}\right]\:=\:\left[{MDA}\right]. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diagonals}\:\mathrm{of} \\ $$$${ABCD}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{diagonal}. \\ $$

Question Number 17233 Answers: 0 Comments: 2

How to find out if cos (cos x)>sin (sin x) cos (sin x)>sin (cos x) cos (sin (cos x))>sin (cos (sin x)) cos (cos (cos x))>sin (sin (sin x)) exam questions. calculators not allowed.

$$\mathrm{How}\:\mathrm{to}\:\mathrm{find}\:\mathrm{out}\:\mathrm{if} \\ $$$$\mathrm{cos}\:\left(\mathrm{cos}\:{x}\right)>\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right) \\ $$$$\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)>\mathrm{sin}\:\left(\mathrm{cos}\:{x}\right) \\ $$$$\mathrm{cos}\:\left(\mathrm{sin}\:\left(\mathrm{cos}\:{x}\right)\right)>\mathrm{sin}\:\left(\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)\right) \\ $$$$\mathrm{cos}\:\left(\mathrm{cos}\:\left(\mathrm{cos}\:{x}\right)\right)>\mathrm{sin}\:\left(\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\right) \\ $$$$\mathrm{exam}\:\mathrm{questions}. \\ $$$$\mathrm{calculators}\:\mathrm{not}\:\mathrm{allowed}. \\ $$

Question Number 16354 Answers: 1 Comments: 0

Question Number 16374 Answers: 1 Comments: 0

The Value of the sum.. Σ_(n=1) ^(13) (i^n +i^(n+1) ), Where i=(√(−1 )) is.. (a.) i (b.) i−1 (c.) −i (d.) 0

$${The}\:{Value}\:{of}\:{the}\:{sum}..\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\left({i}^{{n}} +{i}^{{n}+\mathrm{1}} \right),\:{Where}\:{i}=\sqrt{−\mathrm{1}\:\:} \\ $$$${is}.. \\ $$$$\left({a}.\right)\:{i} \\ $$$$\left({b}.\right)\:{i}−\mathrm{1} \\ $$$$\left({c}.\right)\:−{i} \\ $$$$\left({d}.\right)\:\mathrm{0} \\ $$

Question Number 16061 Answers: 1 Comments: 0

∫ x^5 (√(2 − x^3 )) dx

$$\int\:{x}^{\mathrm{5}} \sqrt{\mathrm{2}\:−\:{x}^{\mathrm{3}} }\:{dx} \\ $$

Question Number 16057 Answers: 0 Comments: 0

please how can i form a differential equation from y=Axϱx

$${please}\:{how}\:{can}\:{i}\:{form}\:{a}\:{differential}\:{equation}\:{from} \\ $$$${y}={Ax}\varrho{x} \\ $$

Question Number 16056 Answers: 0 Comments: 0

y=Axϱ^ x

$${y}={Ax}\hat {\varrho}{x} \\ $$

Question Number 16055 Answers: 1 Comments: 0

Let A′, B′ and C′ be the midpoints of the sides BC, CA and AB of the triangle ABC. Prove that AA′^(→) = (1/2)(AB^(→) + AC^(→) )

$$\mathrm{Let}\:{A}',\:{B}'\:\mathrm{and}\:{C}'\:\mathrm{be}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{BC},\:{CA}\:\mathrm{and}\:{AB}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangle}\:{ABC}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\overset{\rightarrow} {{AA}'}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\overset{\rightarrow} {{AB}}\:+\:\overset{\rightarrow} {{AC}}\right) \\ $$

Question Number 16053 Answers: 0 Comments: 1

number of positive integers a and b and c satisfying a^b^c b^c^a c^a^b =5abc

$${number}\:{of}\:{positive}\:{integers}\:\boldsymbol{{a}}\:\boldsymbol{{and}}\:\boldsymbol{{b}}\:\boldsymbol{{and}}\:\boldsymbol{{c}}\:\boldsymbol{{satisfying}} \\ $$$$\boldsymbol{{a}}^{\boldsymbol{{b}}^{\boldsymbol{{c}}} } \boldsymbol{{b}}^{\boldsymbol{{c}}^{\boldsymbol{{a}}} } \boldsymbol{{c}}^{\boldsymbol{{a}}^{\boldsymbol{{b}}} } =\mathrm{5}\boldsymbol{{abc}} \\ $$

Question Number 16047 Answers: 0 Comments: 1

number of real solution of equation a^(2008) + b^(2008) = 2008ab−2006

$${number}\:{of}\:{real}\:{solution}\:{of}\:{equation}\:\:\:\:{a}^{\mathrm{2008}} \:+\:\:{b}^{\mathrm{2008}} \:\:=\:\mathrm{2008}{ab}−\mathrm{2006}\:\:\:\:\:\:\:\:\: \\ $$

Question Number 16078 Answers: 0 Comments: 0

Question Number 16044 Answers: 1 Comments: 1

A particle is projected horizontally with speed u from point A, which is 10 m above the ground. If the particle hits the inclined plane perpendicularly at point B. [g = 10 m/s^2 ] Find horizontal speed with which the particle was projected.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{horizontally} \\ $$$$\mathrm{with}\:\mathrm{speed}\:{u}\:\mathrm{from}\:\mathrm{point}\:{A},\:\mathrm{which}\:\mathrm{is}\:\mathrm{10} \\ $$$$\mathrm{m}\:\mathrm{above}\:\mathrm{the}\:\mathrm{ground}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{hits} \\ $$$$\mathrm{the}\:\mathrm{inclined}\:\mathrm{plane}\:\mathrm{perpendicularly}\:\mathrm{at} \\ $$$$\mathrm{point}\:{B}.\:\left[{g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right] \\ $$$$\mathrm{Find}\:\mathrm{horizontal}\:\mathrm{speed}\:\mathrm{with}\:\mathrm{which}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{was}\:\mathrm{projected}. \\ $$

Question Number 16079 Answers: 1 Comments: 1

An open flask contains air at 27°C. To what temperature it must be heated to expel one-fourth of the air?

$$\mathrm{An}\:\mathrm{open}\:\mathrm{flask}\:\mathrm{contains}\:\mathrm{air}\:\mathrm{at}\:\mathrm{27}°\mathrm{C}.\:\mathrm{To} \\ $$$$\mathrm{what}\:\mathrm{temperature}\:\mathrm{it}\:\mathrm{must}\:\mathrm{be}\:\mathrm{heated}\:\mathrm{to} \\ $$$$\mathrm{expel}\:\mathrm{one}-\mathrm{fourth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{air}? \\ $$

Question Number 16041 Answers: 1 Comments: 0

If sides a, b, c of ΔABC are in H.P., prove that sin^2 ((A/2)), sin^2 ((B/2)), sin^2 ((C/2)) are in H.P.

$$\mathrm{If}\:\mathrm{sides}\:{a},\:{b},\:{c}\:\mathrm{of}\:\Delta{ABC}\:\mathrm{are}\:\mathrm{in}\:\mathrm{H}.\mathrm{P}., \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{sin}^{\mathrm{2}} \:\left(\frac{{A}}{\mathrm{2}}\right),\:\mathrm{sin}^{\mathrm{2}} \:\left(\frac{{B}}{\mathrm{2}}\right),\:\mathrm{sin}^{\mathrm{2}} \:\left(\frac{{C}}{\mathrm{2}}\right) \\ $$$$\mathrm{are}\:\mathrm{in}\:\mathrm{H}.\mathrm{P}. \\ $$

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