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Guest can you please elaborate?

Alpha particles are energetic (fast) helium nuclei, beta particles are smaller and have half the charge, being energetic electrons (or positrons) only the gamma particles are gamma-ray photons, i.e., they are not massive particles at all, they are a form of electromagnetic radiation, a form more energetic than X-rays.
Source: https://www.quora.com/What-is-the-difference-between-alpha-beta-and-gamma-radiation

0.12 (repetaing)  = 12/99 =   4/33

0.345 (repeating)  = 345/999  = 115/333

So 

4/33  + 115/333  =

[4* 333 + 115 * 33] / [ 33 *333] =    1709 / 3663  =   .466557   (repeating)

So...the sum is  33

Just like the other problem, take the  cube root of 4913   =  17 m

Here you go, NSS....

http://www.kentchemistry.com/links/Nuclear/ArtificalTrans.htm

Volume of sphere = 4/3 x pi x r^3, where r = radius

Radius =3/2 = 1.5 inches

V =4/3 x 3.141592 x 1.5^3

V = ~14 in^3. This is closest to 10^1 =10 - or the 2nd choice on your list.

V_cube = side^3

Therefore

729  = side^3         take the cube root of each side

³√ 729  =  side  =     9 [m]

We have

73.9 * ( 1/2)^ ( t / 3.8)         where t  is the number of days...so....

73.9 * (1/2) ^ (23/ 3.8)  ≈  1.11  g

(1 + tan x  )  / sin x  - sec  x = csc x

(1 + tan x) / sin x  -    1 / cos x

(1 + sin x / cos x) /  sin x  - 1 / cos x

1/sin x + sinx / [ cos x sin x]  - 1/cos x

1 / sin x  + sin x / [ cos x sin x ]  - sin x / [ sinx cos x]

1 / sin x  + sin x [ sin x cos x] -  sin x / [ sin x cos x ]

1 / sin x   =    csc x

E  is true

The function is defined at  x  = 0   as   1

The  limit as  x → 0  exists

F is continuous at  x  = 0    [ the second  part of the piecewise  function "fills in" the "hole' that exists at x = 0  ]

You are still missing NUMBERS from your question. Type them by hand into your question.

yes it is, sorry the asy did not load:

To get from her home to a friend's house, Georgia must go past a school, then a post office:

home---school---post office---freind's house

Signs tell Georgia that the distance she walks from home to the school is  mile, and the distance from the school to the post office is  mile.

The whole trip takes Georgia  minutes. If Georgia can walk a mile in  minutes, then how many miles is the distance from the post office to her friend's house?

Start position ( 0, 0 ):
After 1. Jump, there are 12 different positions with a distance of 5 units

12C2 = 66 x 4 = 264 final positions.

What is the sum of all positive integers that have twice as many digits when written in base $2$ as they have when written in base $3$? Express your answer in base $10$.

2 in base 10 =10 in base 2 = 2 in base 3

8 in base 10 =1000 in base 2 = 22 in base 3.

2 + 8 = 10 the sum of positive integers.

The least common multiple of $1!+2!$, $2!+3!$, $3!+4!$, $4!+5!$, $5!+6!$, $6!+7!$, $7!+8!$, and $8!+9!$ can be expressed in the form $a\cdot b!$, where $a$ and $b$ are integers and $b$ is as large as possible. What is $a+b$?

LCM[1!+2!, 2!+3!, 3!+4!, 4!+5!, 5!+6!, 6!+7!, 7!+8!, 8!+9!] =3,628,800. This can be written as:

1. 10! = 1 + 10 = 11, which is a + b

Is your question missing something?

Verify the following identity:

(tan(x) + 1)/sin(x) - sec(x) = csc(x)

Put (tan(x) + 1)/sin(x) - sec(x) over the common denominator sin(x): (tan(x) + 1)/sin(x) - sec(x) = (1 - sec(x) sin(x) + tan(x))/sin(x):

(1 - sec(x) sin(x) + tan(x))/sin(x) = ^?csc(x)

Multiply both sides by sin(x):

1 - sec(x) sin(x) + tan(x) = ^?csc(x) sin(x)

Write cosecant as 1/sine, secant as 1/cosine and tangent as sine/cosine:

1 - 1/cos(x) sin(x) + sin(x)/cos(x) = ^?1/sin(x) sin(x)

1 - (1/cos(x)) sin(x) + (sin(x)/cos(x)) = 1:

1 = ^?(1/sin(x)) sin(x)

(1/sin(x)) sin(x) = 1:

1 = ^?1

The left hand side and right hand side are identical:

(identity has been verified)

4.

(a^2 + 2ab + 18)^2018

For all exponents n, the number of terms =1/2(n + 2 - 1)(n + 3 - 1). Example: 5th power:

1/2(5 + 2 -1)(5 + 3 - 1) = 1/2 x 6 x 7 = 21 terms. Therefore, to the power of 2018:

1/2 (2018 + 2 - 1)(2018 + 3 - 1) = 1/2 x 2019 x 2020 =2,039,190 number of terms.

I think...

\(f^{-1}\left(\frac{13+2}{5}\right) = f^{-1}(3) =1\)

I am thinking of two numbers in the ratio 5 : 3 . 

The difference between the two numbers is 90. 

What is the sum of the two numbers?

\(\begin{array}{|lrcll|} \hline (1) & \frac{a}{b} &=& \frac{5}{3} \quad & | \quad \cdot b \\ & a &=& \frac53b \\\\ (2) & a-b &=& 90 \quad & | \quad a = \frac53b \\ & \frac53b - b &=& 90 \\ & \frac53b - \frac33 b &=& 90 \\ & \frac23b &=& 90 \quad & | \quad \cdot \frac32 \\ & b &=& 90 \cdot \frac32 \\ & b &=& 3\cdot 45 \\ & b &=& 135 \\ \\ & a &=& \frac53b \quad & | \quad b = 135 \\ & a &=& \frac53 \cdot 135 \\ & a &=& 5\cdot 45 \\ & a &=& 225 \\\\ & a+b &=& 225 + 135 \\ & \mathbf{a+b} & \mathbf{=} & \mathbf{360} \\ \hline \end{array} \)

The sum of the two numbers is 360

Sorry, the asy did not load 

I am thinking of two numbers in the ratio 5 : 3 . The difference between the two numbers is 90. What is the sum of the two numbers?

1)
A frog started from the origin of the coordinate plane and made three jumps.
Each time the frog jumped a distance of 5 units and landed at a point with integer coordinates.
How many different possibilities of the final position of the frog are there?

Startposition ( 0, 0 ):
After 1. Jump, there are 12 different positions with a distance of 5 units.
1.) ( 5, 0 )
2.) ( 4, 3 )
3.) ( 3, 4 )
4.) ( 0, 5 )
5.) ( -3, 4 )
6.) ( -4, 3 )
7.) ( -5, 0 )
8.) ( -4, -3 )
9.) ( -3, -4 )
10.) ( 0, -5 )
11.) ( 3, -4 )
12.) ( 4, -3 )

There are \( 12^3\) = 1728  jumps. But 264 different possibilities of the final position.

1.) (15, 0)

2.) (14, 3)

3.) (13, 4)

4.) (10, 5)

5.) (7, 4)

6.) (6, 3)

7.) (5, 0)

8.) (6, -3)

9.) (7, -4)

10.) (10, -5)

11.) (13, -4)

12.) (14, -3)

13.) (13, 6)

14.) (12, 7)

15.) (9, 8)

16.) (6, 7)

17.) (5, 6)

18.) (4, 3)

19.) (6, -1)

20.) (9, -2)

21.) (12, -1)

22.) (13, 0)

23.) (11, 8)

24.) (8, 9)

25.) (5, 8)

26.) (4, 7)

27.) (3, 4)

28.) (4, 1)

29.) (8, -1)

30.) (11, 0)

31.) (12, 1)

32.) (5, 10)

33.) (2, 9)

34.) (1, 8)

35.) (0, 5)

36.) (1, 2)

37.) (2, 1)

38.) (8, 1)

39.) (9, 2)

40.) (-1, 8)

41.) (-2, 7)

42.) (-3, 4)

43.) (-2, 1)

44.) (-1, 0)

45.) (2, -1)

46.) (6, 1)

47.) (-3, 6)

48.) (-4, 3)

49.) (-3, 0)

50.) (-2, -1)

51.) (1, -2)

52.) (4, -1)

53.) (-5, 0)

54.) (-4, -3)

55.) (-3, -4)

56.) (0, -5)

57.) (3, -4)

58.) (4, -3)

59.) (-3, -6)

60.) (-2, -7)

61.) (1, -8)

62.) (4, -7)

63.) (5, -6)

64.) (-1, -8)

65.) (2, -9)

66.) (5, -8)

67.) (6, -7)

68.) (5, -10)

69.) (8, -9)

70.) (9, -8)

71.) (11, -8)

72.) (12, -7)

73.) (13, -6)

74.) (12, 9)

75.) (11, 10)

76.) (8, 11)

77.) (4, 9)

78.) (3, 6)

79.) (5, 2)

80.) (11, 2)

81.) (12, 3)

82.) (10, 11)

83.) (7, 12)

84.) (4, 11)

85.) (3, 10)

86.) (2, 7)

87.) (7, 2)

88.) (10, 3)

89.) (11, 4)

90.) (4, 13)

91.) (1, 12)

92.) (0, 11)

93.) (1, 4)

94.) (8, 5)

95.) (-2, 11)

96.) (-3, 10)

97.) (-4, 7)

98.) (-2, 3)

99.) (5, 4)

100.) (-4, 9)

101.) (-5, 6)

102.) (-3, 2)

103.) (0, 1)

104.) (3, 2)

105.) (-6, 3)

106.) (-4, -1)

107.) (-1, -2)

108.) (3, 0)

109.) (-2, -5)

110.) (1, -6)

111.) (4, -5)

112.) (5, -4)

113.) (7, -6)

114.) (8, -5)

115.) (11, -4)

116.) (12, -3)

117.) (9, 12)

118.) (6, 13)

119.) (3, 12)

120.) (2, 11)

121.) (2, 5)

122.) (9, 4)

123.) (3, 14)

124.) (0, 13)

125.) (-1, 12)

126.) (-2, 9)

127.) (-1, 6)

128.) (6, 5)

129.) (7, 6)

130.) (-3, 12)

131.) (-4, 11)

132.) (-5, 8)

133.) (-4, 5)

134.) (0, 3)

135.) (4, 5)

136.) (-5, 10)

137.) (-6, 7)

138.) (-5, 4)

139.) (-1, 2)

140.) (2, 3)

141.) (-7, 4)

142.) (-6, 1)

143.) (1, 0)

144.) (-5, -2)

145.) (-1, -4)

146.) (2, -3)

147.) (3, -2)

148.) (3, -6)

149.) (6, -5)

150.) (9, -4)

151.) (10, -3)

152.) (11, -2)

153.) (0, 15)

154.) (-3, 14)

155.) (-4, 13)

156.) (-6, 13)

157.) (-7, 12)

158.) (-8, 9)

159.) (-7, 6)

160.) (-6, 5)

161.) (1, 6)

162.) (-8, 11)

163.) (-9, 8)

164.) (-8, 5)

165.) (-1, 4)

166.) (-10, 5)

167.) (-9, 2)

168.) (-8, 1)

169.) (-8, -1)

170.) (-7, -2)

171.) (0, -1)

172.) (-6, -3)

173.) (0, -3)

174.) (7, -2)

175.) (-9, 12)

176.) (-10, 11)

177.) (-11, 8)

178.) (-9, 4)

179.) (-2, 5)

180.) (-11, 10)

181.) (-12, 7)

182.) (-11, 4)

183.) (-10, 3)

184.) (-7, 2)

185.) (-13, 4)

186.) (-12, 1)

187.) (-11, 0)

188.) (-4, 1)

189.) (-11, -2)

190.) (-10, -3)

191.) (-7, -4)

192.) (-3, -2)

193.) (-9, -4)

194.) (-6, -5)

195.) (-2, -3)

196.) (1, -4)

197.) (5, -2)

198.) (-12, 9)

199.) (-13, 6)

200.) (-12, 3)

201.) (-11, 2)

202.) (-5, 2)

203.) (-14, 3)

204.) (-13, 0)

205.) (-12, -1)

206.) (-9, -2)

207.) (-6, -1)

208.) (-12, -3)

209.) (-11, -4)

210.) (-8, -5)

211.) (-5, -4)

212.) (-10, -5)

213.) (-7, -6)

214.) (-4, -5)

215.) (-4, -7)

216.) (-1, -6)

217.) (2, -5)

218.) (-15, 0)

219.) (-14, -3)

220.) (-13, -4)

221.) (-13, -6)

222.) (-12, -7)

223.) (-9, -8)

224.) (-6, -7)

225.) (-5, -6)

226.) (-11, -8)

227.) (-8, -9)

228.) (-5, -8)

229.) (-5, -10)

230.) (-2, -9)

231.) (2, -7)

232.) (-12, -9)

233.) (-11, -10)

234.) (-8, -11)

235.) (-4, -9)

236.) (-10, -11)

237.) (-7, -12)

238.) (-4, -11)

239.) (-3, -10)

240.) (-4, -13)

241.) (-1, -12)

242.) (0, -11)

243.) (2, -11)

244.) (3, -10)

245.) (4, -9)

246.) (-9, -12)

247.) (-6, -13)

248.) (-3, -12)

249.) (-2, -11)

250.) (-3, -14)

251.) (0, -13)

252.) (1, -12)

253.) (3, -12)

254.) (4, -11)

255.) (0, -15)

256.) (3, -14)

257.) (4, -13)

258.) (6, -13)

259.) (7, -12)

260.) (8, -11)

261.) (9, -12)

262.) (10, -11)

263.) (11, -10)

264.) (12, -9)

Yes you are right. That was very remiss of me. 

I'll hide my erroneous answer for now. While I think on it.

Thanks for pointing that out guest.

Please do not delete your questions.

Somebody might be trying to answer it already!

You can say you do not need an answer any more. That is fine.

But please do not delete the original question. 

What about points like (3,4) melody?

here's an example of a path you didn't consider: the frogs leaps to (3,4) first, then leaps to (3,9) and then back to (3,4)