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Question Number 213459 Answers: 0 Comments: 0

Find tupple natural numbers (a,b,c) such that { ((max{((a+b)/2)+((∣a−b∣)/2) , ((b+c)/2)+((∣b−c∣)/2) ,((c+a)/2)+((∣c−a∣)/2)}=a)),((min{((a+b)/2)−((∣a−b∣)/2) , ((b+c)/2)−((∣b−c∣)/2) , ((c+a)/2)−((∣c−a∣)/2)}=b)) :} where a+b+c = 10

$$\:\:\mathrm{Find}\:\mathrm{tupple}\:\mathrm{natural}\:\mathrm{numbers}\:\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ $$$$\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\begin{cases}{\mathrm{max}\left\{\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}+\frac{\mid\mathrm{a}−\mathrm{b}\mid}{\mathrm{2}}\:,\:\frac{\mathrm{b}+\mathrm{c}}{\mathrm{2}}+\frac{\mid\mathrm{b}−\mathrm{c}\mid}{\mathrm{2}}\:,\frac{\mathrm{c}+\mathrm{a}}{\mathrm{2}}+\frac{\mid\mathrm{c}−\mathrm{a}\mid}{\mathrm{2}}\right\}=\mathrm{a}}\\{\mathrm{min}\left\{\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}−\frac{\mid\mathrm{a}−\mathrm{b}\mid}{\mathrm{2}}\:,\:\frac{\mathrm{b}+\mathrm{c}}{\mathrm{2}}−\frac{\mid\mathrm{b}−\mathrm{c}\mid}{\mathrm{2}}\:,\:\frac{\mathrm{c}+\mathrm{a}}{\mathrm{2}}−\frac{\mid\mathrm{c}−\mathrm{a}\mid}{\mathrm{2}}\right\}=\mathrm{b}}\end{cases} \\ $$$$\:\:\mathrm{where}\:\mathrm{a}+\mathrm{b}+\mathrm{c}\:=\:\mathrm{10} \\ $$

Question Number 213451 Answers: 1 Comments: 0

∫ (1/(z^6 −1)) dz=??

$$\int\:\:\frac{\mathrm{1}}{{z}^{\mathrm{6}} −\mathrm{1}}\:\mathrm{d}{z}=?? \\ $$

Question Number 213439 Answers: 1 Comments: 5

Question Number 213436 Answers: 2 Comments: 0

Question Number 213435 Answers: 0 Comments: 0

A ∈ M_(2×2) ,and ,det (A)≠0 : A^3 = A^2 + A ⇒ det ( A −2I )=?

$$ \\ $$$$\:\:\:\:\:\:\:{A}\:\in\:\mathrm{M}_{\mathrm{2}×\mathrm{2}} \:\:,{and}\:,{det}\:\left({A}\right)\neq\mathrm{0}\::\:\:\:{A}^{\mathrm{3}} \:=\:{A}^{\mathrm{2}} \:+\:{A} \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{det}\:\left(\:{A}\:−\mathrm{2}{I}\:\right)=? \\ $$$$\:\:\:\:\:\: \\ $$

Question Number 213430 Answers: 1 Comments: 0

x,y,z ∈ R { ((x + [y] + {z} = 9,4)),(([x] + {y} + z = 11,3)),(({x} + y + [z] = 10,5)) :} find: x = ?

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{x}\:+\:\left[\mathrm{y}\right]\:+\:\left\{\mathrm{z}\right\}\:=\:\mathrm{9},\mathrm{4}}\\{\left[\mathrm{x}\right]\:+\:\left\{\mathrm{y}\right\}\:+\:\mathrm{z}\:=\:\mathrm{11},\mathrm{3}}\\{\left\{\mathrm{x}\right\}\:+\:\mathrm{y}\:+\:\left[\mathrm{z}\right]\:=\:\mathrm{10},\mathrm{5}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 213428 Answers: 0 Comments: 0

Question Number 213427 Answers: 0 Comments: 0

Question Number 213423 Answers: 2 Comments: 0

⌊ (1/2)x−1⌋ + ⌊ (2/2)x−2⌋+⌊(3/2)x−3⌋+...+⌊((100)/2)x−100⌋ ≤10100 for x non negative integers. find the possible value of x

$$\:\:\lfloor\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}−\mathrm{1}\rfloor\:+\:\lfloor\:\frac{\mathrm{2}}{\mathrm{2}}\mathrm{x}−\mathrm{2}\rfloor+\lfloor\frac{\mathrm{3}}{\mathrm{2}}\mathrm{x}−\mathrm{3}\rfloor+...+\lfloor\frac{\mathrm{100}}{\mathrm{2}}\mathrm{x}−\mathrm{100}\rfloor\:\leqslant\mathrm{10100} \\ $$$$\:\:\mathrm{for}\:\mathrm{x}\:\mathrm{non}\:\mathrm{negative}\:\mathrm{integers}. \\ $$$$\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$

Question Number 213413 Answers: 2 Comments: 0

Find: A= [(√1)] + [(√2)] + [(√3) ]+...+ [(√(323))] = ?

$$\mathrm{Find}: \\ $$$$\mathrm{A}=\:\left[\sqrt{\mathrm{1}}\right]\:+\:\left[\sqrt{\mathrm{2}}\right]\:+\:\left[\sqrt{\mathrm{3}}\:\right]+...+\:\left[\sqrt{\mathrm{323}}\right]\:=\:? \\ $$

Question Number 213417 Answers: 2 Comments: 0

a , b , c , d ∈ N a + b + c + d = 63 Find: maksimum(ab + bc + cd) = ?

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:,\:\mathrm{d}\:\in\:\mathbb{N} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:+\:\mathrm{d}\:=\:\mathrm{63} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{maksimum}\left(\mathrm{ab}\:+\:\mathrm{bc}\:+\:\mathrm{cd}\right)\:=\:? \\ $$

Question Number 213404 Answers: 0 Comments: 3

pls teach me above question ↓↓ (prove real analysis pls) and sorry Mr gaster i cant believe you answer....

$$\mathrm{pls}\:\mathrm{teach}\:\mathrm{me}\:\mathrm{above}\:\mathrm{question} \\ $$$$\downarrow\downarrow\:\left(\mathrm{prove}\:\mathrm{real}\:\mathrm{analysis}\:\mathrm{pls}\right) \\ $$$$\mathrm{and}\:\mathrm{sorry}\:\mathrm{Mr}\:\mathrm{gaster} \\ $$$$\mathrm{i}\:\mathrm{cant}\:\mathrm{believe}\:\mathrm{you}\:\mathrm{answer}.... \\ $$

Question Number 213398 Answers: 2 Comments: 0

One simple Equation pls prove this property Σ_(j=1) ^N a_j ∙Σ_(k=1) ^M b_k =Σ_(j=1) ^N ∙Σ_(k=1) ^M a_j b_k .. and Σ_(j=0) ^N f(a+((b−a)/N)j)∙((b−a)/N)∙Σ_(k=0) ^M g(a+((b−a)/M)k)∙((b−a)/M) =Σ_(j=0) ^N Σ_(k=0) ^M f(a+((b−a)/N)j)g(a+((b−a)/M)k)(((b−a)^2 )/(MN)) But..... that Sum not euqal to ∫_a ^( b) f(z)g(z)dz... why integral form dosen′t work like Summation Σ_(j=1) ^N f(j) ∙Σ_(k=1) ^M g(k)=Σ_(j=1) ^N Σ_(k=1) ^M f(j)g(k) is True But..... ∫_a ^b f(u)du∙ ∫_a ^b g(v)dv isn′t equal to ∫_a ^b f(w)g(w)dw

$$\mathrm{One}\:\mathrm{simple}\:\mathrm{Equation} \\ $$$$\mathrm{pls}\:\mathrm{prove}\:\mathrm{this}\:\mathrm{property} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{a}_{{j}} \centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}{b}_{{k}} =\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:{a}_{{j}} {b}_{{k}} .. \\ $$$$\:\:\mathrm{and} \\ $$$$\underset{{j}=\mathrm{0}} {\overset{{N}} {\sum}}\:{f}\left({a}+\frac{{b}−{a}}{{N}}{j}\right)\centerdot\frac{{b}−{a}}{{N}}\centerdot\underset{{k}=\mathrm{0}} {\overset{{M}} {\sum}}\:\mathrm{g}\left({a}+\frac{{b}−{a}}{{M}}{k}\right)\centerdot\frac{{b}−{a}}{{M}} \\ $$$$=\underset{{j}=\mathrm{0}} {\overset{{N}} {\sum}}\:\underset{{k}=\mathrm{0}} {\overset{{M}} {\sum}}\:{f}\left({a}+\frac{{b}−{a}}{{N}}{j}\right)\mathrm{g}\left({a}+\frac{{b}−{a}}{{M}}{k}\right)\frac{\left({b}−{a}\right)^{\mathrm{2}} }{{MN}} \\ $$$$\mathrm{But}..... \\ $$$$\mathrm{that}\:\mathrm{Sum}\:\mathrm{not}\:\mathrm{euqal}\:\mathrm{to}\:\int_{{a}} ^{\:{b}} \:{f}\left({z}\right)\mathrm{g}\left({z}\right)\mathrm{d}{z}... \\ $$$$\mathrm{why}\:\mathrm{integral}\:\mathrm{form}\:\mathrm{dosen}'\mathrm{t}\:\mathrm{work} \\ $$$$\:\mathrm{like}\:\mathrm{Summation} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{f}\left({j}\right)\:\centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:\mathrm{g}\left({k}\right)=\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:{f}\left({j}\right)\mathrm{g}\left({k}\right) \\ $$$$\:\mathrm{is}\:\mathrm{True}\:\mathrm{But}..... \\ $$$$\int_{{a}} ^{{b}} \:{f}\left({u}\right)\mathrm{d}{u}\centerdot\:\int_{{a}} ^{{b}} \:\mathrm{g}\left({v}\right)\mathrm{d}{v}\:\:\mathrm{isn}'\mathrm{t}\:\mathrm{equal}\:\mathrm{to}\: \\ $$$$\int_{{a}} ^{{b}} \:{f}\left({w}\right)\mathrm{g}\left({w}\right)\mathrm{d}{w} \\ $$

Question Number 213397 Answers: 1 Comments: 0

Question Number 213391 Answers: 1 Comments: 0

Solve the system of equations where a,b,c≥0 a−2bc=b−2ac=c−2ab a+b+c=2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}\:\mathrm{where}\:{a},{b},{c}\geqslant\mathrm{0} \\ $$$${a}−\mathrm{2}{bc}={b}−\mathrm{2}{ac}={c}−\mathrm{2}{ab} \\ $$$${a}+{b}+{c}=\mathrm{2}\: \\ $$

Question Number 213330 Answers: 1 Comments: 1

Question Number 213323 Answers: 1 Comments: 1

Question Number 213322 Answers: 4 Comments: 1

Question Number 213298 Answers: 1 Comments: 0

Question 29. Theres 2 same-ratio sequences {a_n },{b_n } with a ratio of non-zero And if two sums of each sequences (Σ_(n=1) ^∞ a_n ,Σ_(n=1) ^∞ b_n ) are convergent, and two equations Σ_(n=1) ^∞ a_n b_n =(Σ_(n=1) ^∞ a_n )×(Σ_(n=1) ^∞ b_n ) and 3×Σ_(n=1) ^∞ ∣a_(2n) ∣=7×Σ_(n=1) ^∞ ∣a_(3n) ∣ are true. If Σ_(n=1) ^∞ ((b_(2n−1) +b_(3n+1) )/b_n )=S, then Find the value of 120S. (korea university exam question)

Question Number 213292 Answers: 0 Comments: 0

hey tinku tara I cant plot functions even i am logged in it says check if the variable name is “x” and you are logged in

$${hey}\:{tinku}\:{tara} \\ $$$${I}\:{cant}\:{plot}\:{functions} \\ $$$${even}\:{i}\:{am}\:{logged}\:{in} \\ $$$${it}\:{says}\:{check}\:{if}\:{the}\:{variable}\:{name}\:{is}\:``{x}''\:{and}\:{you}\:{are}\:{logged}\:{in} \\ $$

Question Number 213291 Answers: 1 Comments: 0

Find domain of y_(213291) : y_(213291) =((3+e^((x^2 −3x+2)/(x−6)) )/(log_(3/4) (√(x^2 −(1/4)))))

$${Find}\:{domain}\:{of}\:{y}_{\mathrm{213291}} : \\ $$$${y}_{\mathrm{213291}} =\frac{\mathrm{3}+{e}^{\frac{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}{{x}−\mathrm{6}}} }{\mathrm{log}_{\frac{\mathrm{3}}{\mathrm{4}}} \sqrt{{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}} \\ $$

Question Number 213290 Answers: 1 Comments: 0

for every real set R , f∈R and f is Smooth function. and f is f∈C^2 ∀_x f^((1)) (x)>0 , f^((2)) (x)<0 then prove ∣∫_0 ^( t) cos(f(x))dx∣≤(2/(f^((1)) (t))) t∈R

$$\mathrm{for}\:\mathrm{every}\:\mathrm{real}\:\mathrm{set}\:\mathbb{R}\:,\:{f}\in\mathbb{R} \\ $$$$\mathrm{and}\:{f}\:\mathrm{is}\:\mathrm{Smooth}\:\mathrm{function}.\:\mathrm{and}\:{f}\:\mathrm{is}\:{f}\in\mathcal{C}^{\mathrm{2}} \\ $$$$\forall_{{x}} \:{f}^{\left(\mathrm{1}\right)} \left({x}\right)>\mathrm{0}\:,\:{f}^{\left(\mathrm{2}\right)} \left({x}\right)<\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mid\int_{\mathrm{0}} ^{\:{t}} \:\mathrm{cos}\left({f}\left({x}\right)\right)\mathrm{d}{x}\mid\leq\frac{\mathrm{2}}{{f}^{\left(\mathrm{1}\right)} \left({t}\right)} \\ $$$${t}\in\mathbb{R} \\ $$

Question Number 213288 Answers: 0 Comments: 0

Question Number 213283 Answers: 1 Comments: 0

a_h is Cauchy Sequence. Sequence {a_h }_(h=1) ^n Satisfy Σ_(h=1) ^n a_h =0 , Σ_(h=1) ^n a_h ^2 =1 find minimum value of Summation a_1 a_n +Σ_(h=1) ^(n−1) a_h a_(h+1) (korea university math contest problem)

$${a}_{{h}} \:\mathrm{is}\:\mathrm{Cauchy}\:\mathrm{Sequence}. \\ $$$$\mathrm{Sequence}\:\left\{{a}_{{h}} \right\}_{{h}=\mathrm{1}} ^{{n}} \mathrm{Satisfy}\:\underset{{h}=\mathrm{1}} {\overset{{n}} {\sum}}\:{a}_{{h}} =\mathrm{0}\:,\:\underset{{h}=\mathrm{1}} {\overset{{n}} {\sum}}\:{a}_{{h}} ^{\mathrm{2}} =\mathrm{1} \\ $$$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{Summation} \\ $$$$\:{a}_{\mathrm{1}} {a}_{{n}} +\underset{{h}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\:{a}_{{h}} {a}_{{h}+\mathrm{1}} \\ $$$$\left(\mathrm{korea}\:\mathrm{university}\:\mathrm{math}\:\mathrm{contest}\:\mathrm{problem}\right) \\ $$

Question Number 213281 Answers: 2 Comments: 1

Question Number 213276 Answers: 1 Comments: 2

y^2 = − 4px At (−(1/3),1)→ 1= −4p (− (1/3) − h) At (−(5/3),2)→ 4 = −4p (− (5/3) − h) 4 = ((− (5/3) − h)/(− (1/3) − h)) (4/3) + 4h = (5/3) + h 3h = (1/3) h = (1/9)

$$\mathrm{y}^{\mathrm{2}} \:=\:−\:\mathrm{4px} \\ $$$$\:\mathrm{At}\:\left(−\frac{\mathrm{1}}{\mathrm{3}},\mathrm{1}\right)\rightarrow\:\mathrm{1}=\:−\mathrm{4p}\:\left(−\:\frac{\mathrm{1}}{\mathrm{3}}\:−\:\mathrm{h}\right) \\ $$$$\:\mathrm{At}\:\left(−\frac{\mathrm{5}}{\mathrm{3}},\mathrm{2}\right)\rightarrow\:\mathrm{4}\:=\:−\mathrm{4p}\:\left(−\:\frac{\mathrm{5}}{\mathrm{3}}\:−\:\mathrm{h}\right) \\ $$$$\:\:\:\:\:\mathrm{4}\:=\:\frac{−\:\frac{\mathrm{5}}{\mathrm{3}}\:−\:\mathrm{h}}{−\:\frac{\mathrm{1}}{\mathrm{3}}\:−\:\mathrm{h}} \\ $$$$\:\:\:\:\frac{\mathrm{4}}{\mathrm{3}}\:+\:\mathrm{4h}\:=\:\frac{\mathrm{5}}{\mathrm{3}}\:+\:\mathrm{h}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3h}\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{h}\:=\:\frac{\mathrm{1}}{\mathrm{9}} \\ $$

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