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Question Number 19264 Answers: 1 Comments: 2

Question Number 19250 Answers: 0 Comments: 2

Why arg(z) + arg(z^ ) = 2kπ, k ∈ Z? Shouldn′t it be always 0?

$$\mathrm{Why}\:\mathrm{arg}\left({z}\right)\:+\:\mathrm{arg}\left(\bar {{z}}\right)\:=\:\mathrm{2}{k}\pi,\:{k}\:\in\:{Z}? \\ $$$$\mathrm{Shouldn}'\mathrm{t}\:\mathrm{it}\:\mathrm{be}\:\boldsymbol{\mathrm{always}}\:\mathrm{0}? \\ $$

Question Number 19247 Answers: 1 Comments: 2

Question Number 20046 Answers: 0 Comments: 3

Anyone who has a very high passion for Mathematics should join AoPS community. It has all types of groups, from middle school math to olympiad to college math. But a new user can only post 3 topics. This restriction will be removed after 2 weeks. Thanks for joining. Sign up on: artofproblemsolving.com For question forum: artofproblemsolving.com/community

$$\mathrm{Anyone}\:\mathrm{who}\:\mathrm{has}\:\mathrm{a}\:\mathrm{very}\:\mathrm{high}\:\mathrm{passion} \\ $$$$\mathrm{for}\:\mathrm{Mathematics}\:\mathrm{should}\:\mathrm{join}\:\mathrm{AoPS} \\ $$$$\mathrm{community}.\:\mathrm{It}\:\mathrm{has}\:\mathrm{all}\:\mathrm{types}\:\mathrm{of}\:\mathrm{groups}, \\ $$$$\mathrm{from}\:\mathrm{middle}\:\mathrm{school}\:\mathrm{math}\:\mathrm{to}\:\mathrm{olympiad} \\ $$$$\mathrm{to}\:\mathrm{college}\:\mathrm{math}.\:\mathrm{But}\:\mathrm{a}\:\mathrm{new}\:\mathrm{user}\:\mathrm{can} \\ $$$$\mathrm{only}\:\mathrm{post}\:\mathrm{3}\:\mathrm{topics}.\:\mathrm{This}\:\mathrm{restriction}\:\mathrm{will} \\ $$$$\mathrm{be}\:\mathrm{removed}\:\mathrm{after}\:\mathrm{2}\:\mathrm{weeks}.\:\mathrm{Thanks}\:\mathrm{for} \\ $$$$\mathrm{joining}.\:\mathrm{Sign}\:\mathrm{up}\:\mathrm{on}: \\ $$$$\mathrm{artofproblemsolving}.\mathrm{com} \\ $$$$\mathrm{For}\:\mathrm{question}\:\mathrm{forum}: \\ $$$$\mathrm{artofproblemsolving}.\mathrm{com}/\mathrm{community} \\ $$

Question Number 19245 Answers: 1 Comments: 1

Let f(x) be a quadratic polynomial with integer coefficients such that f(0) and f(1) are odd integers. Prove that the equation f(x) = 0 does not have an integer solution.

$$\mathrm{Let}\:{f}\left({x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{quadratic}\:\mathrm{polynomial} \\ $$$$\mathrm{with}\:\mathrm{integer}\:\mathrm{coefficients}\:\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{0}\right) \\ $$$$\mathrm{and}\:{f}\left(\mathrm{1}\right)\:\mathrm{are}\:\mathrm{odd}\:\mathrm{integers}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{equation}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{does}\:\mathrm{not}\:\mathrm{have}\:\mathrm{an} \\ $$$$\mathrm{integer}\:\mathrm{solution}. \\ $$

Question Number 19243 Answers: 0 Comments: 3

Let a, b, c be the sides opposite the angles A, B and C respectively of a ΔABC. Find the value of k such that (a) a + b = kc (b) cot (A/2) + cot (B/2) = k cot (C/2).

$$\mathrm{Let}\:{a},\:{b},\:{c}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{opposite}\:\mathrm{the} \\ $$$$\mathrm{angles}\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{respectively}\:\mathrm{of}\:\mathrm{a} \\ $$$$\Delta\mathrm{ABC}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{a}\right)\:{a}\:+\:{b}\:=\:{kc} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\:+\:\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:=\:{k}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}. \\ $$

Question Number 19239 Answers: 1 Comments: 0

Assume that a, b, c and d are positive integers such that a^5 = b^4 , c^3 = d^2 and c − a = 19. Determine d − b.

$$\mathrm{Assume}\:\mathrm{that}\:{a},\:{b},\:{c}\:\mathrm{and}\:{d}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:{a}^{\mathrm{5}} \:=\:{b}^{\mathrm{4}} ,\:{c}^{\mathrm{3}} \:=\:{d}^{\mathrm{2}} \:\mathrm{and} \\ $$$${c}\:−\:{a}\:=\:\mathrm{19}.\:\mathrm{Determine}\:{d}\:−\:{b}. \\ $$

Question Number 19230 Answers: 1 Comments: 0

Question Number 19223 Answers: 1 Comments: 7

A particle P is sliding down a frictionless hemispherical bowl. It passes the point A at t = 0. At this instant of time, the horizontal component of its velocity is v. A bead Q of the same mass as P is ejected from A at t = 0 along the horizontal direction, with the speed v. Friction between the bead and the string may be neglected. Let t_P and t_Q be the respective times taken by P and Q to reach the point B. Then (a) t_P < t_Q (b) t_P = t_Q (c) t_P > t_Q (d) (t_P /t_Q ) = ((length of at arc ACB)/(length of chord AB))

$$\mathrm{A}\:\mathrm{particle}\:{P}\:\mathrm{is}\:\mathrm{sliding}\:\mathrm{down}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{hemispherical}\:\mathrm{bowl}.\:\mathrm{It}\:\mathrm{passes}\:\mathrm{the}\:\mathrm{point} \\ $$$${A}\:\mathrm{at}\:{t}\:=\:\mathrm{0}.\:\mathrm{At}\:\mathrm{this}\:\mathrm{instant}\:\mathrm{of}\:\mathrm{time},\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{component}\:\mathrm{of}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{is} \\ $$$${v}.\:\mathrm{A}\:\mathrm{bead}\:{Q}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{mass}\:\mathrm{as}\:{P}\:\mathrm{is} \\ $$$$\mathrm{ejected}\:\mathrm{from}\:{A}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{along}\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{direction},\:\mathrm{with}\:\mathrm{the}\:\mathrm{speed}\:{v}. \\ $$$$\mathrm{Friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{bead}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{string}\:\mathrm{may}\:\mathrm{be}\:\mathrm{neglected}.\:\mathrm{Let}\:{t}_{{P}} \:\mathrm{and}\:{t}_{{Q}} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{respective}\:\mathrm{times}\:\mathrm{taken}\:\mathrm{by}\:{P}\:\mathrm{and} \\ $$$${Q}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{point}\:{B}.\:\mathrm{Then} \\ $$$$\left({a}\right)\:{t}_{{P}} \:<\:{t}_{{Q}} \\ $$$$\left({b}\right)\:{t}_{{P}} \:=\:{t}_{{Q}} \\ $$$$\left({c}\right)\:{t}_{{P}} \:>\:{t}_{{Q}} \\ $$$$\left({d}\right)\:\frac{{t}_{{P}} }{{t}_{{Q}} }\:=\:\frac{\mathrm{length}\:\mathrm{of}\:\mathrm{at}\:\mathrm{arc}\:{ACB}}{\mathrm{length}\:\mathrm{of}\:\mathrm{chord}\:{AB}} \\ $$

Question Number 19215 Answers: 1 Comments: 0

STATEMENT-1 : For every natural number n ≥ 2, (1/(√1)) + (1/(√2)) + ..... (1/(√n)) > (√n) and STATEMENT-2 : For every natural number n ≥ 2, (√(n(n + 1))) < n + 1

$$\mathrm{STATEMENT}-\mathrm{1}\::\:\mathrm{For}\:\mathrm{every}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\geqslant\:\mathrm{2},\:\frac{\mathrm{1}}{\sqrt{\mathrm{1}}}\:+\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:+\:.....\:\frac{\mathrm{1}}{\sqrt{{n}}}\:>\:\sqrt{{n}} \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\mathrm{For}\:\mathrm{every}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\geqslant\:\mathrm{2},\:\sqrt{{n}\left({n}\:+\:\mathrm{1}\right)}\:<\:{n}\:+\:\mathrm{1} \\ $$

Question Number 19204 Answers: 0 Comments: 1

If L= [((1 0 0)),((3 1 0)),((2 4 1)) ]and B= [(3),(2),(1) ] x_1 =−2 ; x_2 =1 ; x_3 =5 find U (numerical analysis)

$$\mathrm{If}\:\mathrm{L}=\begin{bmatrix}{\mathrm{1}\:\:\mathrm{0}\:\:\mathrm{0}}\\{\mathrm{3}\:\:\mathrm{1}\:\:\mathrm{0}}\\{\mathrm{2}\:\:\mathrm{4}\:\:\mathrm{1}}\end{bmatrix}\mathrm{and}\:\mathrm{B}=\begin{bmatrix}{\mathrm{3}}\\{\mathrm{2}}\\{\mathrm{1}}\end{bmatrix} \\ $$$$\mathrm{x}_{\mathrm{1}} =−\mathrm{2}\:;\:\mathrm{x}_{\mathrm{2}} =\mathrm{1}\:;\:\mathrm{x}_{\mathrm{3}} =\mathrm{5} \\ $$$$\mathrm{find}\:\mathrm{U} \\ $$$$\left(\mathrm{numerical}\:\mathrm{analysis}\right) \\ $$$$ \\ $$

Question Number 19198 Answers: 1 Comments: 7

Find all integer solutions of the system: 35x + 63y + 45z = 1, ∣x∣ < 9, ∣y∣ < 5, ∣z∣ < 7.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system}: \\ $$$$\mathrm{35}{x}\:+\:\mathrm{63}{y}\:+\:\mathrm{45}{z}\:=\:\mathrm{1},\:\mid{x}\mid\:<\:\mathrm{9},\:\mid{y}\mid\:<\:\mathrm{5}, \\ $$$$\mid{z}\mid\:<\:\mathrm{7}. \\ $$

Question Number 19195 Answers: 1 Comments: 1

∫((5x^4 +4x^5 )/((x^5 +x+1)^2 )) solve the intgration

$$\int\frac{\mathrm{5}{x}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{5}} }{\left({x}^{\mathrm{5}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{solve}\:{the}\:{intgration} \\ $$

Question Number 19194 Answers: 2 Comments: 0

Question Number 19193 Answers: 1 Comments: 0

The sum of two positive integers is 52 and their LCM is 168. Find the numbers.

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{is}\:\mathrm{52} \\ $$$$\mathrm{and}\:\mathrm{their}\:\mathrm{LCM}\:\mathrm{is}\:\mathrm{168}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{numbers}. \\ $$

Question Number 19192 Answers: 0 Comments: 2

Find a natural number ′n′ such that 3^9 + 3^(12) + 3^(15) + 3^n is a perfect cube of an integer.

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:'\mathrm{n}'\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{3}^{\mathrm{9}} \:+\:\mathrm{3}^{\mathrm{12}} \:+\:\mathrm{3}^{\mathrm{15}} \:+\:\mathrm{3}^{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{cube}\:\mathrm{of} \\ $$$$\mathrm{an}\:\mathrm{integer}. \\ $$

Question Number 19262 Answers: 1 Comments: 0

Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p ×{x−q)×p}+q×{x−r)×q} + r×{x−p)×r}=0, then x is given by

Question Number 19182 Answers: 2 Comments: 0

Find all three digit numbers abc (with a ≠ 0) such that a^2 + b^2 + c^2 , is divisible by 26.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}\:\mathrm{digit}\:\mathrm{numbers}\:{abc}\:\left(\mathrm{with}\right. \\ $$$$\left.\mathrm{a}\:\neq\:\mathrm{0}\right)\:\mathrm{such}\:\mathrm{that}\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} ,\:\mathrm{is}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{26}. \\ $$

Question Number 19171 Answers: 1 Comments: 1

log_(√2) (√(2(√(2(√(2(√(2 ))))))))

$${log}_{\sqrt{\mathrm{2}}} \:\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}\:\:\:\:}}}} \\ $$

Question Number 19154 Answers: 0 Comments: 0

If f(x) = (9^x /(9^x + 3)), find the value f((1/(2007))) + f((2/(2007))) + f((3/(2007))) + ... + f(((2006)/(2006)))

$$\mathrm{If}\:{f}\left({x}\right)\:=\:\frac{\mathrm{9}^{{x}} }{\mathrm{9}^{{x}} \:+\:\mathrm{3}},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2007}}\right)\:+\:{f}\left(\frac{\mathrm{2}}{\mathrm{2007}}\right)\:+\:{f}\left(\frac{\mathrm{3}}{\mathrm{2007}}\right)\:+\:...\:+\:{f}\left(\frac{\mathrm{2006}}{\mathrm{2006}}\right) \\ $$

Question Number 19150 Answers: 1 Comments: 4

A semicircle is tangent to both legs of a right triangle and has its centre on the hypotenuse. The hypotenuse is partitioned into 4 segments, with lengths 3, 12, 12, and x, as shown in the figure. Determine the value of ′x′.

$$\mathrm{A}\:\mathrm{semicircle}\:\mathrm{is}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{both}\:\mathrm{legs}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{right}\:\mathrm{triangle}\:\mathrm{and}\:\mathrm{has}\:\mathrm{its}\:\mathrm{centre}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{hypotenuse}.\:\mathrm{The}\:\mathrm{hypotenuse}\:\mathrm{is} \\ $$$$\mathrm{partitioned}\:\mathrm{into}\:\mathrm{4}\:\mathrm{segments},\:\mathrm{with}\:\mathrm{lengths} \\ $$$$\mathrm{3},\:\mathrm{12},\:\mathrm{12},\:\mathrm{and}\:{x},\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{figure}. \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:'{x}'. \\ $$

Question Number 19148 Answers: 1 Comments: 0

If f(x)= determinant (((sin x+sin2x+sin 3x),(sin 2x),(sin 3x)),(( 3+4 sin x),( 3),(4 sin x)),(( 1+sin x),( sin x),( 1))) then the value of ∫_( 0) ^(π/2) f(x) dx is

$$\mathrm{If} \\ $$$${f}\left({x}\right)=\begin{vmatrix}{\mathrm{sin}\:{x}+\mathrm{sin2}{x}+\mathrm{sin}\:\mathrm{3}{x}}&{\mathrm{sin}\:\mathrm{2}{x}}&{\mathrm{sin}\:\mathrm{3}{x}}\\{\:\:\:\:\:\:\:\:\mathrm{3}+\mathrm{4}\:\mathrm{sin}\:{x}}&{\:\:\:\:\mathrm{3}}&{\mathrm{4}\:\mathrm{sin}\:{x}}\\{\:\:\:\:\:\:\:\:\:\:\mathrm{1}+\mathrm{sin}\:{x}}&{\:\mathrm{sin}\:{x}}&{\:\:\:\:\mathrm{1}}\end{vmatrix} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:\:\:\mathrm{is} \\ $$

Question Number 19140 Answers: 0 Comments: 9

A racing car travels on a track (without banking) ABCDEFA. ABC is a circular arc of radius 2R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is μ = 0.1. The maximum speed of the car is 50 ms^(−1) . Find the minimum time for completing one round.

$$\mathrm{A}\:\mathrm{racing}\:\mathrm{car}\:\mathrm{travels}\:\mathrm{on}\:\mathrm{a}\:\mathrm{track}\:\left(\mathrm{without}\right. \\ $$$$\left.\mathrm{banking}\right)\:{ABCDEFA}.\:{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{arc}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{2}{R}.\:{CD}\:\mathrm{and}\:{FA}\:\mathrm{are} \\ $$$$\mathrm{straight}\:\mathrm{paths}\:\mathrm{of}\:\mathrm{length}\:{R}\:\mathrm{and}\:{DEF}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{circular}\:\mathrm{arc}\:\mathrm{of}\:\mathrm{radius}\:{R}\:=\:\mathrm{100}\:\mathrm{m}.\:\mathrm{The} \\ $$$$\mathrm{co}-\mathrm{efficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{on}\:\mathrm{the}\:\mathrm{road}\:\mathrm{is}\:\mu\:= \\ $$$$\mathrm{0}.\mathrm{1}.\:\mathrm{The}\:\mathrm{maximum}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{is} \\ $$$$\mathrm{50}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{time}\:\mathrm{for} \\ $$$$\mathrm{completing}\:\mathrm{one}\:\mathrm{round}. \\ $$

Question Number 19137 Answers: 1 Comments: 1

Figure shows (x, t), (y, t) diagram of a particle moving in 2-dimensions. If the particle has a mass of 500 g, find the force (direction and magnitude) acting on the particle.

$$\mathrm{Figure}\:\mathrm{shows}\:\left({x},\:{t}\right),\:\left({y},\:{t}\right)\:\mathrm{diagram}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{particle}\:\mathrm{moving}\:\mathrm{in}\:\mathrm{2}-\mathrm{dimensions}.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{has}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{500}\:\mathrm{g},\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{force}\:\left(\mathrm{direction}\:\mathrm{and}\:\mathrm{magnitude}\right)\:\mathrm{acting} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{particle}. \\ $$

Question Number 19135 Answers: 1 Comments: 0

solve for x: 2^(∣x+2∣) −∣2^(x+1) −1∣=2^(x+1) +1

$${solve}\:{for}\:{x}: \\ $$$$\mathrm{2}^{\mid{x}+\mathrm{2}\mid} −\mid\mathrm{2}^{{x}+\mathrm{1}} −\mathrm{1}\mid=\mathrm{2}^{{x}+\mathrm{1}} +\mathrm{1} \\ $$

Question Number 19134 Answers: 1 Comments: 0

If (1/((243)^x )) = (729)^y = 3^3 , then find the value of 5x + 6y.

$$\mathrm{If}\:\frac{\mathrm{1}}{\left(\mathrm{243}\right)^{{x}} }\:=\:\left(\mathrm{729}\right)^{{y}} =\:\mathrm{3}^{\mathrm{3}} ,\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{5}{x}\:+\:\mathrm{6}{y}. \\ $$

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