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Question Number 110592 Answers: 3 Comments: 0

How many pairs of integers x and y satisfy the equation (1/x)+(1/y)=(1/(32))

$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{32}} \\ $$

Question Number 110591 Answers: 3 Comments: 0

Find the sum of all positive two−digit integers that are divisible by each of their digits.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive} \\ $$$$\mathrm{two}−\mathrm{digit}\:\mathrm{integers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{each}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}. \\ $$

Question Number 110673 Answers: 2 Comments: 0

Let k be a real number such that the inequality (√(x−3))+(√(6−x))≥k has a solution. Find the maximum value of k.

$$ \\ $$$$\mathrm{Let}\:\mathrm{k}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{inequality}\:\sqrt{\mathrm{x}−\mathrm{3}}+\sqrt{\mathrm{6}−\mathrm{x}}\geqslant\mathrm{k}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{solution}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{k}. \\ $$

Question Number 110587 Answers: 1 Comments: 0

A palindrome is a number that remains the same when its numbers are reversed. The number n and n+192 are three−digit and four−digit palindromes respectively. What is the sum of the digits of m? Can this be solved mathematically?

$$\mathrm{A}\:\mathrm{palindrome}\:\mathrm{is}\:\mathrm{a}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{remains}\:\mathrm{the}\:\mathrm{same}\:\mathrm{when}\:\mathrm{its}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{reversed}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{n}\:\mathrm{and} \\ $$$$\mathrm{n}+\mathrm{192}\:\mathrm{are}\:\mathrm{three}−\mathrm{digit}\:\mathrm{and} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{palindromes}\:\mathrm{respectively}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{m}?\: \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{mathematically}? \\ $$

Question Number 110586 Answers: 1 Comments: 0

The length of the interval of solution a≤2x+5≤b is 15. What is b−a?

$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{solution} \\ $$$$\mathrm{a}\leqslant\mathrm{2x}+\mathrm{5}\leqslant\mathrm{b}\:\mathrm{is}\:\mathrm{15}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{b}−\mathrm{a}? \\ $$

Question Number 110725 Answers: 2 Comments: 0

x^x^x^x^(...) =2 x=?

$${x}^{{x}^{{x}^{{x}^{...} } } } =\mathrm{2}\:\:\:\:\:\:\:{x}=? \\ $$

Question Number 110562 Answers: 2 Comments: 0

Let x and y be integers such that xy≠1, x^2 ≠y and y^2 ≠x. (i) Show that p∣xy−1 and p∣x^2 −y then p∣y^2 −x where p is a prime. (ii) Let p be a prime. Suppose that p∣x^2 −y and p∣y^2 −x, must p∣xy−1? [If yes, then prove it. If no, then give a counter example]

$$\mathrm{Let}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{be}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{xy}\neq\mathrm{1},\:\mathrm{x}^{\mathrm{2}} \neq\mathrm{y}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} \neq\mathrm{x}. \\ $$$$ \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{p}\mid\mathrm{xy}−\mathrm{1}\:\mathrm{and}\:\mathrm{p}\mid\mathrm{x}^{\mathrm{2}} −\mathrm{y} \\ $$$$\mathrm{then}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x}\:\mathrm{where}\:\mathrm{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{a}\:\mathrm{prime}.\:\mathrm{Suppose}\:\mathrm{that} \\ $$$$\mathrm{p}\mid\mathrm{x}^{\mathrm{2}} −\mathrm{y}\:\mathrm{and}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x},\:\mathrm{must}\:\mathrm{p}\mid\mathrm{xy}−\mathrm{1}? \\ $$$$ \\ $$$$\left[\mathrm{If}\:\mathrm{yes},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{it}.\:\mathrm{If}\:\mathrm{no},\:\mathrm{then}\:\mathrm{give}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{counter}\:\mathrm{example}\right] \\ $$

Question Number 110565 Answers: 0 Comments: 15

Let n∈N. Using the formula lcm(a,b) = ((ab)/(gcd(a,b))) and lcm(a,b,c) =lcm(lcm(a,b),c), find all the possible value of ((6•lcm(n,n+1,n+2,n+3))/(n(n+1)(n+2)(n+3)))

$$\mathrm{Let}\:\mathrm{n}\in\mathbb{N}.\:\mathrm{Using}\:\mathrm{the}\:\mathrm{formula}\:\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right) \\ $$$$=\:\frac{\mathrm{ab}}{\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right)}\:\mathrm{and}\:\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ $$$$=\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right),\:\mathrm{find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{6}\bullet\mathrm{lcm}\left(\mathrm{n},\mathrm{n}+\mathrm{1},\mathrm{n}+\mathrm{2},\mathrm{n}+\mathrm{3}\right)}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)\left(\mathrm{n}+\mathrm{3}\right)} \\ $$

Question Number 110688 Answers: 0 Comments: 0

Let a,b and c be positive integers such that ab+1∣bc+1 and bc+1∣ca+1. Show that ab+1 is the sum of two squares.

$$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$

Question Number 110551 Answers: 2 Comments: 0

Question Number 110550 Answers: 0 Comments: 0

Question Number 110549 Answers: 2 Comments: 1

Question Number 110545 Answers: 1 Comments: 2

Question Number 110543 Answers: 1 Comments: 0

Question Number 113557 Answers: 1 Comments: 0

If the numerically smaller root of x^2 +mx=2 is more than the other one, find the value of m.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{numerically}\:\mathrm{smaller}\:\mathrm{root}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{2}} +{mx}=\mathrm{2}\:\mathrm{is}\:\mathrm{more}\:\mathrm{than}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}. \\ $$

Question Number 113644 Answers: 1 Comments: 0

Question Number 110539 Answers: 0 Comments: 4

it seems too hard for many here to post their questions as questions, answers as answers and comments as comments... hereby I introduce the next step: I′ll post answers and the task is, find questions to these answers (1) ζ(3)+πln2 (2) πH_0 (7) (3) true ∀x∈C\Q (4) _2 F_1 ((3/2), (1/5), (2/3), sin^(−1) ((x+1)/(x−1)))

$${it}\:{seems}\:{too}\:{hard}\:{for}\:{many}\:{here}\:{to}\:{post} \\ $$$${their}\:{questions}\:{as}\:{questions},\:{answers}\:{as} \\ $$$${answers}\:{and}\:{comments}\:{as}\:{comments}... \\ $$$${hereby}\:{I}\:{introduce}\:{the}\:{next}\:{step}:\:{I}'{ll}\:{post} \\ $$$${answers}\:{and}\:{the}\:{task}\:{is},\:{find}\:{questions} \\ $$$${to}\:{these}\:{answers} \\ $$$$\left(\mathrm{1}\right)\:\zeta\left(\mathrm{3}\right)+\pi{ln}\mathrm{2} \\ $$$$\left(\mathrm{2}\right)\:\pi{H}_{\mathrm{0}} \left(\mathrm{7}\right) \\ $$$$\left(\mathrm{3}\right)\:{true}\:\forall{x}\in\mathbb{C}\backslash\mathbb{Q} \\ $$$$\left(\mathrm{4}\right)\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{2}},\:\frac{\mathrm{1}}{\mathrm{5}},\:\frac{\mathrm{2}}{\mathrm{3}},\:{sin}^{−\mathrm{1}} \frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right) \\ $$

Question Number 110644 Answers: 1 Comments: 1

The Diophantine equation x^2 +y^2 +1 =N(xy+1) has infinitely many integer solutions if N equals? Any help please?

$$\mathrm{The}\:\mathrm{Diophantine}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}\:=\mathrm{N}\left(\mathrm{xy}+\mathrm{1}\right)\:\mathrm{has} \\ $$$$\mathrm{infinitely}\:\mathrm{many}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{if}\:\mathrm{N}\:\mathrm{equals}? \\ $$$$\mathrm{Any}\:\mathrm{help}\:\mathrm{please}? \\ $$

Question Number 110888 Answers: 3 Comments: 0

....calculus.... please solve : Ω_1 =∫_0 ^( (π/4)) ((√(tan(x))) +(√(cot(x))) )dx=?? Ω_2 =∫_0 ^(π/4) tan(x)ln((1+tan^2 (x)))dx =?? ...M.N.july 1970#... Good luck

$$\:\:\:\:\:\:\:\:\:\:....{calculus}.... \\ $$$${please}\:{solve}\:: \\ $$$$ \\ $$$$\Omega_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left(\sqrt{{tan}\left({x}\right)}\:+\sqrt{{cot}\left({x}\right)}\:\right){dx}=?? \\ $$$$\:\Omega_{\mathrm{2}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {tan}\left({x}\right){ln}\left(\left(\mathrm{1}+{tan}^{\mathrm{2}} \left({x}\right)\right)\right){dx}\:=?? \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:...\mathscr{M}.\mathscr{N}.{july}\:\mathrm{1970}#... \\ $$$$\:\mathscr{G}{ood}\:{luck} \\ $$$$ \\ $$$$ \\ $$

Question Number 110887 Answers: 1 Comments: 2

Question Number 110528 Answers: 1 Comments: 0

Question Number 110525 Answers: 0 Comments: 1

Question Number 110524 Answers: 1 Comments: 0

A blind man is to place 6 letters into 6 pigeon holes, how many ways can atleast 5 letters be wrongly placed? (Note that only one letter must be in a pigeon hole).

$$\mathrm{A}\:\mathrm{blind}\:\mathrm{man}\:\mathrm{is}\:\mathrm{to}\:\mathrm{place}\:\mathrm{6}\:\mathrm{letters}\:\mathrm{into}\:\mathrm{6} \\ $$$$\mathrm{pigeon}\:\mathrm{holes},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can} \\ $$$$\mathrm{atleast}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{be}\:\mathrm{wrongly}\:\mathrm{placed}? \\ $$$$\left(\mathrm{Note}\:\mathrm{that}\:\mathrm{only}\:\mathrm{one}\:\mathrm{letter}\:\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{pigeon}\:\mathrm{hole}\right). \\ $$

Question Number 110523 Answers: 0 Comments: 0

In the square PQRS, K is the midpoint of PQ, L is the midpoint of QR, M is the midpoint RS, N is the midpoint of SP and O is the midpoint of KM. A line segment is drawn from each pair of points from (K,L,M,N,O,P,Q,R,S). These line segments create points of intersections not contained in (K,L,M,N,O,P,Q,R,S). How many distinct such points are there?

$$ \\ $$$$\mathrm{In}\:\mathrm{the}\:\mathrm{square}\:\mathrm{PQRS},\:\mathrm{K}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{PQ},\:\mathrm{L}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{QR},\:\mathrm{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{RS},\:\mathrm{N}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{SP}\:\mathrm{and}\:\mathrm{O}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint} \\ $$$$\mathrm{of}\:\mathrm{KM}.\:\mathrm{A}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{from} \\ $$$$\mathrm{each}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{points}\:\mathrm{from} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{These}\:\mathrm{line} \\ $$$$\mathrm{segments}\:\mathrm{create}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{intersections}\:\mathrm{not}\:\mathrm{contained}\:\mathrm{in} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{distinct}\:\mathrm{such}\:\mathrm{points}\:\mathrm{are}\:\mathrm{there}? \\ $$

Question Number 110510 Answers: 0 Comments: 0

Question Number 110519 Answers: 1 Comments: 2

17x ≡ 3 (mod 29)

$$\:\:\:\:\:\:\mathrm{17x}\:\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{29}\right) \\ $$

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