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Question Number 212522 Answers: 3 Comments: 0

in how many ways we can distribute 6 distinct balls in 3 identical boxes

$$\:\:\boldsymbol{{in}}\:\boldsymbol{{how}}\:\boldsymbol{{many}}\:\boldsymbol{{ways}}\:\boldsymbol{{we}} \\ $$$$\boldsymbol{{can}}\:\boldsymbol{{distribute}}\:\mathrm{6}\:\boldsymbol{{distinct}} \\ $$$$\boldsymbol{{balls}}\:\boldsymbol{{in}}\:\mathrm{3}\:\boldsymbol{{identical}}\:\boldsymbol{{boxes}} \\ $$

Question Number 212519 Answers: 1 Comments: 0

certificate: (x−1)∣(x^(2n+1) −1)and(x+1)∣(x^(2n+1) +1) All established [2024.10.16]

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\left({x}−\mathrm{1}\right)\mid\left({x}^{\mathrm{2}{n}+\mathrm{1}} −\mathrm{1}\right)\mathrm{and}\left({x}+\mathrm{1}\right)\mid\left({x}^{\mathrm{2}{n}+\mathrm{1}} +\mathrm{1}\right) \\ $$$$\mathrm{All}\:\mathrm{established} \\ $$$$\left[\mathrm{2024}.\mathrm{10}.\mathrm{16}\right] \\ $$

Question Number 212518 Answers: 2 Comments: 0

⇃

$$\:\:\:\cancel{\underbrace{\downharpoonleft}\underline{}\:} \\ $$

Question Number 212516 Answers: 0 Comments: 0

Question Number 212515 Answers: 1 Comments: 1

The numbers of pairs of natural numbers (x,y) with x,y ≤ 33 that satisfy 5 ∣ 3^x^(y−1) + 2^y^(2x) is ... (A) 295 (B) 296 (C) 297 (D) 298 (E) 299

$$\:\:\mathrm{The}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{natural}\: \\ $$$$\:\:\:\mathrm{numbers}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{with}\:\mathrm{x},\mathrm{y}\:\leqslant\:\mathrm{33}\:\mathrm{that}\: \\ $$$$\:\:\:\mathrm{satisfy}\:\mathrm{5}\:\mid\:\mathrm{3}^{\mathrm{x}^{\mathrm{y}−\mathrm{1}} } \:+\:\mathrm{2}^{\mathrm{y}^{\mathrm{2x}} } \:\mathrm{is}\:...\: \\ $$$$\:\:\left(\mathrm{A}\right)\:\mathrm{295}\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{296}\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{297}\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{298}\:\:\:\left(\mathrm{E}\right)\:\mathrm{299} \\ $$

Question Number 212514 Answers: 1 Comments: 0

Question Number 212513 Answers: 1 Comments: 0

certificate: x^(2n) −1 Can always be x+1 be divisible

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:{x}^{\mathrm{2}{n}} −\mathrm{1} \\ $$$$\mathrm{Can}\:\mathrm{always}\:\mathrm{be}\:{x}+\mathrm{1}\:\mathrm{be}\:\mathrm{divisible}\:\:\:\:\:\:\:\:\: \\ $$

Question Number 212508 Answers: 1 Comments: 1

Question Number 212506 Answers: 1 Comments: 0

Find: ((− ((18)/(1 + i (√3)))))^(1/4)

$$\mathrm{Find}:\:\:\:\:\:\:\sqrt[{\mathrm{4}}]{−\:\frac{\mathrm{18}}{\mathrm{1}\:+\:\boldsymbol{\mathrm{i}}\:\sqrt{\mathrm{3}}}} \\ $$

Question Number 212503 Answers: 0 Comments: 0

Question Number 212500 Answers: 1 Comments: 0

Can we exactly find r_(max) (a). r^2 +((2rt)/a)(t+2(√(ar)))=t^2 ∀ t is parameter

$${Can}\:{we}\:{exactly}\:{find}\:{r}_{{max}} \left({a}\right). \\ $$$${r}^{\mathrm{2}} +\frac{\mathrm{2}{rt}}{{a}}\left({t}+\mathrm{2}\sqrt{{ar}}\right)={t}^{\mathrm{2}} \:\:\:\:\forall\:{t}\:{is}\:{parameter} \\ $$

Question Number 212499 Answers: 2 Comments: 0

Given a,b,c and d are reals numbers such that { ((a^2 +b^2 =10)),((c^2 +d^2 =10 )),((ab+cd=0)) :} Find ac + bd.

$$\:\:\mathrm{Given}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{and}\:\mathrm{d}\:\mathrm{are}\:\mathrm{reals}\: \\ $$$$\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\begin{cases}{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\mathrm{10}}\\{\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} =\mathrm{10}\:}\\{\mathrm{ab}+\mathrm{cd}=\mathrm{0}}\end{cases} \\ $$$$\:\:\mathrm{Find}\:\mathrm{ac}\:+\:\mathrm{bd}. \\ $$

Question Number 212498 Answers: 0 Comments: 0

Question Number 212497 Answers: 0 Comments: 0

Question Number 212496 Answers: 0 Comments: 0

If 1.1!+3.2!+...+(2n−1).n! = a.(n+1)!+b(1!+2!+...+(n+1)!)+c for a,b,c integers number find 2a−b+3c

$$\:\mathrm{If}\:\mathrm{1}.\mathrm{1}!+\mathrm{3}.\mathrm{2}!+...+\left(\mathrm{2n}−\mathrm{1}\right).\mathrm{n}!\: \\ $$$$\:=\:\mathrm{a}.\left(\mathrm{n}+\mathrm{1}\right)!+\mathrm{b}\left(\mathrm{1}!+\mathrm{2}!+...+\left(\mathrm{n}+\mathrm{1}\right)!\right)+\mathrm{c}\: \\ $$$$\:\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{integers}\:\mathrm{number} \\ $$$$\:\mathrm{find}\:\mathrm{2a}−\mathrm{b}+\mathrm{3c}\: \\ $$

Question Number 212495 Answers: 0 Comments: 0

Question Number 212492 Answers: 0 Comments: 0

Given that x ∈ R( (the set of real numbers)d an n ∈ N^∗ (the set of positive naturalu nmbers) prove that : ∣cos x∣+∣cos 2x∣+∣cos 3x∣+…+∣cos(n+1)x∣≥(n/2)

$$\mathrm{Given}\:\mathrm{that}\:{x}\:\in\:\mathbb{R}\left(\right. \\ $$$$\:\left(\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{real}\:\mathrm{numbers}\right)\mathrm{d} \\ $$$$\mathrm{an}\:{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$\left(\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{naturalu}\right. \\ $$$$\left.\mathrm{nmbers}\right)\:\mathrm{prove}\:\mathrm{that}\:: \\ $$$$\mid\mathrm{cos}\:{x}\mid+\mid\mathrm{cos}\:\mathrm{2}{x}\mid+\mid\mathrm{cos}\:\mathrm{3}{x}\mid+\ldots+\mid\mathrm{cos}\left({n}+\mathrm{1}\right){x}\mid\geq\frac{{n}}{\mathrm{2}} \\ $$

Question Number 212479 Answers: 1 Comments: 0

{ ((2x + 3y − z = 7)),((4x − y + 2z = 1)),((−x + 5y + 3z = 14)) :} find: x,y,z = ?

$$\begin{cases}{\mathrm{2x}\:+\:\mathrm{3y}\:−\:\mathrm{z}\:=\:\mathrm{7}}\\{\mathrm{4x}\:−\:\mathrm{y}\:+\:\mathrm{2z}\:=\:\mathrm{1}}\\{−\mathrm{x}\:+\:\mathrm{5y}\:+\:\mathrm{3z}\:=\:\mathrm{14}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\mathrm{x},\mathrm{y},\mathrm{z}\:=\:? \\ $$

Question Number 212477 Answers: 3 Comments: 0

Question Number 212470 Answers: 1 Comments: 0

Question Number 212467 Answers: 1 Comments: 0

Use the integration by parts. ∫ (2/( (√(r^2 −4)))) dr

$$\mathrm{Use}\:\:\mathrm{the}\:\:\mathrm{integration}\:\:\mathrm{by}\:\:\mathrm{parts}. \\ $$$$ \\ $$$$\int\:\frac{\mathrm{2}}{\:\sqrt{{r}^{\mathrm{2}} −\mathrm{4}}}\:{dr}\: \\ $$

Question Number 212464 Answers: 1 Comments: 1

Question Number 212462 Answers: 0 Comments: 1

1. In a plane there are nt poins arranged such that nostraight line passesg throuh more than 3 points.h Wat is the maximume numbr of straight lines thatcan pass throughy exactl 3 points2. If thee numbr 3 in the aboveo questin is changed to anys poitive integer greater than2 denoted as x whatu wold the answer be

$$\mathrm{1}.\:\mathrm{In}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{there}\:\mathrm{are}\:\mathrm{nt} \\ $$$$\mathrm{poins}\:\mathrm{arranged}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{nostraight}\:\mathrm{line}\:\mathrm{passesg} \\ $$$$\mathrm{throuh}\:\mathrm{more}\:\mathrm{than}\:\mathrm{3}\:\mathrm{points}.\mathrm{h} \\ $$$$\mathrm{Wat}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximume} \\ $$$$\mathrm{numbr}\:\mathrm{of}\:\mathrm{straight}\:\mathrm{lines}\: \\ $$$$\mathrm{thatcan}\:\mathrm{pass}\:\mathrm{throughy} \\ $$$$\mathrm{exactl}\:\mathrm{3}\:\mathrm{points2}.\:\mathrm{If}\:\mathrm{thee} \\ $$$$\mathrm{numbr}\:\mathrm{3}\:\mathrm{in}\:\mathrm{the}\:\mathrm{aboveo} \\ $$$$\mathrm{questin}\:\mathrm{is}\:\mathrm{changed}\:\mathrm{to}\:\mathrm{anys} \\ $$$$\mathrm{poitive}\:\mathrm{integer}\:\mathrm{greater}\: \\ $$$$\mathrm{than2}\:\mathrm{denoted}\:\mathrm{as}\:\mathrm{x}\:\mathrm{whatu} \\ $$$$\mathrm{wold}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{be} \\ $$

Question Number 212461 Answers: 0 Comments: 1

prove: ∫_0 ^(+∞) (dx/( (√(x^4 +25x^2 +160))))=∫_0 ^(+∞) (dx/( (√(x^4 −95x^2 +2560))))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{prove}: \\ $$$$\:\:\int_{\mathrm{0}} ^{+\infty} \frac{{dx}}{\:\sqrt{{x}^{\mathrm{4}} +\mathrm{25}{x}^{\mathrm{2}} +\mathrm{160}}}=\int_{\mathrm{0}} ^{+\infty} \frac{{dx}}{\:\sqrt{{x}^{\mathrm{4}} −\mathrm{95}{x}^{\mathrm{2}} +\mathrm{2560}}} \\ $$

Question Number 212460 Answers: 0 Comments: 0

Question Number 212448 Answers: 3 Comments: 0

If x^2 − x + 3 = 0 Find x + (9/x^2 ) = ?

$$\mathrm{If}\:\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{x}\:+\:\mathrm{3}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\:\mathrm{x}\:+\:\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }\:=\:? \\ $$

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