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Question Number 221078 Answers: 0 Comments: 0

There is a widely circulated number game among the people called "Making Zero". The rules of a simplified version of the game are as follows: Two players, A and B, each extend one hand. At the beginning, both players use this hand to represent a number (each number is an integer from 1 to 9 and can be chosen arbitrarily) as the initial state. Then, one side is chosen as the first player, who uses his hand to touch the other player's hand. The number represented by his hand is changed to the sum of "the number originally represented by his hand" and "the number represented by the other player's hand", and only the units digit is kept as the new number, while the other player's number remains unchanged. Subsequently, the other side repeats this process. Generally, the game continues until one side "makes zero", at which point that side is declared the winner. Specifically, if after several rounds, neither side can "make zero", that is, the situation enters a "dead loop", we call this situation a "singular situation". For example, if the initial states of A and B are "2" and "4" respectively, and A is the first player, after one round, A and B become "6" and "4", and after another round, they become "6" and "0". At this point, B "makes zero" and wins. We can represent any situation formed by both sides at any moment in the game as an ordered array (a, b), which means: at this moment, the number represented by the first player's hand is a, and the number represented by the second player's hand is b. For example, in the previous example, the game process can be represented as (2, 4) → (6, 4) → (6, 0), with the second player winning. Or, in another game, it can be represented as (5, 3) → (8, 3) → (8, 1) → (9, 1) → (9, 0), with the second player winning. For any situation (a, b), what conditions should a and b meet for the first player to win; what conditions should a and b meet for the second player to win; what conditions should a and b meet for (a, b) to be a "singular situation"? (Analyze the relationship between a and b)

$$ \\ $$There is a widely circulated number game among the people called "Making Zero". The rules of a simplified version of the game are as follows: Two players, A and B, each extend one hand. At the beginning, both players use this hand to represent a number (each number is an integer from 1 to 9 and can be chosen arbitrarily) as the initial state. Then, one side is chosen as the first player, who uses his hand to touch the other player's hand. The number represented by his hand is changed to the sum of "the number originally represented by his hand" and "the number represented by the other player's hand", and only the units digit is kept as the new number, while the other player's number remains unchanged. Subsequently, the other side repeats this process. Generally, the game continues until one side "makes zero", at which point that side is declared the winner. Specifically, if after several rounds, neither side can "make zero", that is, the situation enters a "dead loop", we call this situation a "singular situation". For example, if the initial states of A and B are "2" and "4" respectively, and A is the first player, after one round, A and B become "6" and "4", and after another round, they become "6" and "0". At this point, B "makes zero" and wins. We can represent any situation formed by both sides at any moment in the game as an ordered array (a, b), which means: at this moment, the number represented by the first player's hand is a, and the number represented by the second player's hand is b. For example, in the previous example, the game process can be represented as (2, 4) → (6, 4) → (6, 0), with the second player winning. Or, in another game, it can be represented as (5, 3) → (8, 3) → (8, 1) → (9, 1) → (9, 0), with the second player winning. For any situation (a, b), what conditions should a and b meet for the first player to win; what conditions should a and b meet for the second player to win; what conditions should a and b meet for (a, b) to be a "singular situation"? (Analyze the relationship between a and b)

Question Number 221060 Answers: 1 Comments: 0

Please what’s TSA of a frustrum of a cone? Any easy method?

Question Number 221057 Answers: 2 Comments: 1

Question Number 221056 Answers: 1 Comments: 0

Question Number 221052 Answers: 2 Comments: 0

Question Number 221050 Answers: 0 Comments: 1

Question Number 221049 Answers: 0 Comments: 0

∫ ((cos x)/((x + cos x)^2 )) + ((2(1 − sin x)^2 )/((x + cos x)^3 )) dx

$$ \\ $$$$\:\:\:\:\int\:\:\frac{\mathrm{cos}\:{x}}{\left({x}\:+\:\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{2}\left(\mathrm{1}\:−\:\mathrm{sin}\:{x}\right)^{\mathrm{2}} }{\left({x}\:+\:\mathrm{cos}\:{x}\right)^{\mathrm{3}} }\:\:\mathrm{d}{x}\: \\ $$$$\: \\ $$

Question Number 221048 Answers: 2 Comments: 0

∫_0 ^(π/2) cosec (x−(π/3))cosec (x−(π/6))dx

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{cosec}\:\left({x}−\frac{\pi}{\mathrm{3}}\right)\mathrm{cosec}\:\left({x}−\frac{\pi}{\mathrm{6}}\right){dx}\: \\ $$

Question Number 221047 Answers: 1 Comments: 0

Question Number 221044 Answers: 0 Comments: 1

If V be a function of x and y, prove that (∂^2 V/∂x^2 )+(∂^2 V/∂y^2 )=(∂^2 V/∂r^2 )+(1/r) (∂V/∂r)+(1/r^2 ) (∂^2 V/∂θ^2 ), where x=r cos θ , y=rsin θ

$${If}\:{V}\:{be}\:{a}\:{function}\:{of}\:{x}\:{and}\:{y},\:{prove}\:{that} \\ $$$$\frac{\partial^{\mathrm{2}} {V}}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {V}}{\partial{y}^{\mathrm{2}} }=\frac{\partial^{\mathrm{2}} {V}}{\partial{r}^{\mathrm{2}} }+\frac{\mathrm{1}}{{r}}\:\frac{\partial{V}}{\partial{r}}+\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\:\frac{\partial^{\mathrm{2}} {V}}{\partial\theta^{\mathrm{2}} }, \\ $$$${where}\:{x}={r}\:\mathrm{cos}\:\theta\:,\:{y}={r}\mathrm{sin}\:\theta \\ $$

Question Number 221036 Answers: 0 Comments: 0

prove ∫_(P∈[ε,ε+δ]) (√((((d )/dt)f(t))^2 +(((d )/dt)g(t))^2 )) dt≤∫_(P∈[ε,ε+δ]) ((C_1 f^((1)) (t)+C_2 g^((1)) (t))/( (√(C_1 ^2 +C_2 ^2 )))) dt

$$\mathrm{prove} \\ $$$$\int_{{P}\in\left[\epsilon,\epsilon+\delta\right]} \sqrt{\left(\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}{f}\left({t}\right)\right)^{\mathrm{2}} +\left(\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\mathrm{g}\left({t}\right)\right)^{\mathrm{2}} }\:\mathrm{d}{t}\leq\int_{{P}\in\left[\epsilon,\epsilon+\delta\right]} \:\frac{{C}_{\mathrm{1}} {f}^{\left(\mathrm{1}\right)} \left({t}\right)+{C}_{\mathrm{2}} \mathrm{g}^{\left(\mathrm{1}\right)} \left({t}\right)}{\:\sqrt{{C}_{\mathrm{1}} ^{\mathrm{2}} +{C}_{\mathrm{2}} ^{\mathrm{2}} }}\:\mathrm{d}{t} \\ $$

Question Number 221025 Answers: 1 Comments: 0

Question Number 221017 Answers: 1 Comments: 0

Question Number 221034 Answers: 1 Comments: 0

Question Number 221012 Answers: 3 Comments: 4

Question Number 221008 Answers: 0 Comments: 0

Question Number 220995 Answers: 2 Comments: 0

Question Number 220987 Answers: 1 Comments: 0

Let a,b,c be positive reals such that abc=1.prove that (1/(a^3 (b+c)))+(1/(b^3 (c+a)))+(1/(c^3 (a+b)))≥(3/2)

$${Let}\:{a},{b},{c}\:{be}\:{positive}\:{reals}\:{such}\:{that}\:{abc}=\mathrm{1}.{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} \left({b}+{c}\right)}+\frac{\mathrm{1}}{{b}^{\mathrm{3}} \left({c}+{a}\right)}+\frac{\mathrm{1}}{{c}^{\mathrm{3}} \left({a}+{b}\right)}\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 220973 Answers: 1 Comments: 0

Prove:Σ_(k=0) ^n (−(1/3))^k cos^3 (3^(k−n) π)=(3/4)[(−(1/3))^(n+1) +cos(π/3^n )]

$$\mathrm{Prove}:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{k}} \mathrm{cos}^{\mathrm{3}} \left(\mathrm{3}^{{k}−{n}} \pi\right)=\frac{\mathrm{3}}{\mathrm{4}}\left[\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{n}+\mathrm{1}} +\mathrm{cos}\frac{\pi}{\mathrm{3}^{{n}} }\right] \\ $$

Question Number 220972 Answers: 1 Comments: 2

∫_(−∞) ^(+∞) ∫_(−∞) ^(+∞) (1/((x^2 +y^2 +4)^2 )) dxdy x=rcos(θ) y=rsin(θ) ∣∣J∣∣=rdrdθ ∫_0 ^( 2π) ∫_0 ^( ∞) (r/((r^2 +4)^2 ))drdθ=(1/8)∫_0 ^( 2π) dθ=(π/4) Q. if ∫_0 ^( ∞) ∫_0 ^( ∞) (1/((x^2 +y^2 +4)^2 )) dxdy 0≤r<∞ , 0≤θ≤(π/2).....??? I really confuse how could i select interval of integral

$$\int_{−\infty} ^{+\infty} \int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }\:\mathrm{d}{x}\mathrm{d}{y} \\ $$$${x}={r}\mathrm{cos}\left(\theta\right) \\ $$$${y}={r}\mathrm{sin}\left(\theta\right) \\ $$$$\mid\mid\boldsymbol{{J}}\mid\mid={r}\mathrm{d}{r}\mathrm{d}\theta \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \int_{\mathrm{0}} ^{\:\infty} \:\:\:\frac{{r}}{\left({r}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }\mathrm{d}{r}\mathrm{d}\theta=\frac{\mathrm{1}}{\mathrm{8}}\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\mathrm{d}\theta=\frac{\pi}{\mathrm{4}} \\ $$$${Q}.\:\mathrm{if}\:\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }\:\mathrm{d}{x}\mathrm{d}{y} \\ $$$$\mathrm{0}\leq{r}<\infty\:,\:\mathrm{0}\leq\theta\leq\frac{\pi}{\mathrm{2}}.....??? \\ $$$$\mathrm{I}\:\mathrm{really}\:\mathrm{confuse}\:\mathrm{how}\:\mathrm{could}\:\:\mathrm{i}\:\mathrm{select}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{integral} \\ $$

Question Number 220971 Answers: 1 Comments: 6

Question Number 221680 Answers: 2 Comments: 4

I suspect π=i∫_i ^(−1) (((z−1)dz)/( z(√(z^2 +1)))) someone please help confirm or reject!

$${I}\:{suspect} \\ $$$$\pi={i}\underset{\boldsymbol{{i}}} {\overset{−\mathrm{1}} {\int}}\frac{\left({z}−\mathrm{1}\right){dz}}{\:{z}\sqrt{{z}^{\mathrm{2}} +\mathrm{1}}} \\ $$$${someone}\:{please}\:{help}\:{confirm}\:{or}\:{reject}! \\ $$

Question Number 220968 Answers: 0 Comments: 0

Question Number 220964 Answers: 0 Comments: 0

Find the general solution of the differential equation x^2 (d^3 y/dx^3 ) + x(d^2 y/dx^2 )−6(dy/dx)+6(y/x)=((x ln x+1)/x^2 ),[x>0]

$${Find}\:{the}\:{general}\:{solution}\:{of}\:{the}\:{differential}\:{equation} \\ $$$${x}^{\mathrm{2}} \:\frac{{d}^{\mathrm{3}} {y}}{{dx}^{\mathrm{3}} }\:+\:{x}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{6}\frac{{dy}}{{dx}}+\mathrm{6}\frac{{y}}{{x}}=\frac{{x}\:\mathrm{ln}\:{x}+\mathrm{1}}{{x}^{\mathrm{2}} },\left[{x}>\mathrm{0}\right] \\ $$

Question Number 220963 Answers: 1 Comments: 0

Let f:R^2 →R be defined by f(x,y)={(y/(sin y_( 1, y=0) )), y≠0 Then the integral (1/π^2 )∫_(x=0) ^1 ∫_(y=sin^(−1) x) ^(π/2) f(x,y)dy dx correct upto three decimal places,is...

$${Let}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}\:{be}\:{defined}\:{by}\:{f}\left({x},{y}\right)=\left\{\frac{{y}}{\underset{\:\:\mathrm{1},\:{y}=\mathrm{0}} {\mathrm{sin}\:{y}}},\:{y}\neq\mathrm{0}\right. \\ $$$${Then}\:{the}\:{integral}\:\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{{y}=\mathrm{sin}^{−\mathrm{1}} {x}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{f}\left({x},{y}\right){dy}\:{dx}\:{correct}\:{upto}\:{three}\:{decimal}\:{places},{is}... \\ $$

Question Number 220958 Answers: 1 Comments: 1

for x, y, z >0 find the maximum of x^m y^n z^k subject to ax+by+cz=d.

$${for}\:{x},\:{y},\:{z}\:>\mathrm{0}\:{find}\:{the}\:{maximum}\:{of} \\ $$$${x}^{{m}} {y}^{{n}} {z}^{{k}} \:{subject}\:{to}\:{ax}+{by}+{cz}={d}. \\ $$

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