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Question Number 174599 Answers: 0 Comments: 0

Question Number 174594 Answers: 1 Comments: 0

Let σ(n) be the sum of all positive divisors of the integer n and let p be any prime number. show that σ(n)<2n holds true for all n of the form n=p^2 . Mastermind

$$\mathrm{Let}\:\sigma\left(\mathrm{n}\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive} \\ $$$$\mathrm{divisors}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{and}\:\mathrm{let}\:\mathrm{p}\:\mathrm{be} \\ $$$$\mathrm{any}\:\mathrm{prime}\:\mathrm{number}.\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\sigma\left(\mathrm{n}\right)<\mathrm{2n}\:\mathrm{holds}\:\mathrm{true}\:\mathrm{for}\:\mathrm{all}\:\mathrm{n}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{form}\:\mathrm{n}=\mathrm{p}^{\mathrm{2}} . \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Question Number 174630 Answers: 1 Comments: 0

lim_(n→∞) Σ_(r=1) ^∞ (r/(n^2 +r))

$$\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{r}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{r}}{{n}^{\mathrm{2}} +{r}} \\ $$

Question Number 174591 Answers: 1 Comments: 0

The drawing below shows two equilateral triangles with side length a. The triangle are horizontally shifted by (a/2). Find the intersection area A of the two triangles (grey area).

$$\mathrm{The}\:\mathrm{drawing}\:\mathrm{below}\:\mathrm{shows}\:\mathrm{two}\: \\ $$$$\mathrm{equilateral}\:\mathrm{triangles}\:\mathrm{with}\:\mathrm{side}\:\mathrm{length} \\ $$$$\boldsymbol{\mathrm{a}}.\:\mathrm{The}\:\mathrm{triangle}\:\mathrm{are}\:\mathrm{horizontally}\:\mathrm{shifted} \\ $$$$\mathrm{by}\:\frac{\mathrm{a}}{\mathrm{2}}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{area}\:\mathrm{A}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{triangles}\:\left(\mathrm{grey}\:\mathrm{area}\right). \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 174587 Answers: 0 Comments: 2

Question Number 174583 Answers: 0 Comments: 0

Your friend makes 6 field goals and 2 free throws. You make twice as many field goals as your friend and half the number of free throws. How many points do you have. explain the order of operation you followed

$$\:\mathrm{Your}\:\mathrm{friend}\:\mathrm{makes}\:\mathrm{6}\:\mathrm{field}\:\mathrm{goals}\:\mathrm{and}\: \\ $$$$\:\:\mathrm{2}\:\mathrm{free}\:\mathrm{throws}.\:\mathrm{You}\:\mathrm{make}\:\mathrm{twice}\:\mathrm{as} \\ $$$$\:\mathrm{many}\:\mathrm{field}\:\mathrm{goals}\:\mathrm{as}\:\mathrm{your}\:\mathrm{friend}\:\mathrm{and}\: \\ $$$$\:\mathrm{half}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{free}\:\mathrm{throws}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{points}\:\mathrm{do}\:\mathrm{you}\:\mathrm{have}.\:\mathrm{explain}\:\mathrm{the} \\ $$$$\mathrm{order}\:\mathrm{of}\:\mathrm{operation}\:\mathrm{you}\:\mathrm{followed} \\ $$$$\:\:\:\:\: \\ $$

Question Number 174582 Answers: 0 Comments: 0

Your friend makes 4 field goals . you make three times as many field goals as your friend plus one field goals. how many points do you have. Explain the order of operation you followed

$$\:\mathrm{Your}\:\mathrm{friend}\:\mathrm{makes}\:\mathrm{4}\:\mathrm{field}\:\mathrm{goals}\:.\:\mathrm{you} \\ $$$$\:\mathrm{make}\:\mathrm{three}\:\mathrm{times}\:\mathrm{as}\:\mathrm{many}\:\mathrm{field}\:\mathrm{goals} \\ $$$$\mathrm{as}\:\mathrm{your}\:\mathrm{friend}\:\mathrm{plus}\:\mathrm{one}\:\mathrm{field}\:\mathrm{goals}. \\ $$$$\:\mathrm{how}\:\mathrm{many}\:\mathrm{points}\:\mathrm{do}\:\mathrm{you}\:\mathrm{have}.\: \\ $$$$\mathrm{Explain}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{operation}\:\mathrm{you}\: \\ $$$$\mathrm{followed} \\ $$

Question Number 174581 Answers: 1 Comments: 0

3^3^3^3 find last two digits

$$\mathrm{3}^{\mathrm{3}^{\mathrm{3}^{\mathrm{3}} } } \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{last}}\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{digits}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 174579 Answers: 0 Comments: 0

find minimum value of: ((a^2 (b−1)^2 +2ab+(a−1)^2 )/(a(ab+1))) [all∈R]

$$\:\:\:\:{find}\:{minimum}\:{value}\:{of}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{{a}}^{\mathrm{2}} \left(\boldsymbol{{b}}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{2}\boldsymbol{{ab}}+\left(\boldsymbol{{a}}−\mathrm{1}\right)^{\mathrm{2}} }{\boldsymbol{{a}}\left(\boldsymbol{{ab}}+\mathrm{1}\right)}\:\:\:\:\:\:\:\:\:\left[\boldsymbol{{all}}\in\boldsymbol{{R}}\right] \\ $$

Question Number 174578 Answers: 0 Comments: 0