Thanks sir

(x, y)=(1, 2); (2, 1)

(X^2+Y^2+XY)/(2XY)=7/4 (X^3+Y^3)/(2XY)=9/4 X=2 Y=1 final answer

(10^11)+2ab(10^11)=22 (10^5^6)+2ab(105^6) (2^5^5^6)+2ab(2^55^6) (1^1^1^3^2)+1ab(1^1^13^2) (1^1)+ab(3^2) (3^2) (ab ➖ 3ab+2).

b=(44-a^2)/(2a+1) is natural only when a=0, 2, 3, corresponding to b= 44, 8, 5.

Consider 1^x, 16^x, 81^x, 625^x 1^x=(1^x)⁴ 16^x=(2⁴)^x --> 16^x=(2^x)⁴ Similarly: 81^x=(3^x)⁴ 625^x=(5^x)⁴ AM=¼[(1^x)⁴+(2^x)⁴+(3^x)⁴+(5^x)⁴] N=4AM D=4(30^x) N/D=AM/(30^x) Note that 30=1(2)3(5) Thus 30^x=[1(2)3(5)]^x =(1^x)(2^x)(3^x)(5^x) =GM N/D=1 --> AM=GM meaning 1^x=16^x=81^x=625^x. Therefore x=0 AM=¼[(1^x)⁴+(2^x)⁴+(3^x)⁴+(5^x)⁴] GM=[{(1^x)(2^x)(3^x)(4x)}⁴]^¼ =(1^x)(2^x)(3^x)(5^x)

Writing a = 5 ^x , b = 3 ^ x, c = 2 ^ x and d = 1 ^ x one gets a^4 + b^4 + c ^ 4 + d ^ 4 = 4 a b c d ( a ^ 2 - b ^2) ^ 2 + ( c ^ 2 - d ^2) ^ 2 + 2 (a ^2 * b^2 + c^2 * d ^2 - 2 a b c d) = 0 ( a ^ 2 - b ^2) ^ 2 + ( c ^ 2 - d ^2) ^ 2 + 2 ( a b - c d ) ^ 2 = 0 For real a, b, c d we get a = b, and c = d (5 /2) ^ x = 1 i.e x = 0

X is 37.

AM-GM inequality: (1 + a^4 + b^4 + c^4)/4 >= abc --> a = b = c = 1 --> x = 0

1^x+4^4^x+9^9^x5^4^x/5^6^x+5^6^x+5^6x+5^6^x 1^x+2^2^2^2^x+3^2^3^2^x+1^2^2^x/1^3^2^x+1^3^2^x+1^3^2x+1^3^2^x 1^1^1^1^x+1^1^1^1^x+1^1^x/1^1^x+1^1^x+1^1^x+3^2^x (x ➖ 3x+2).

Nice

Writing 2^ x = a one gets a^4 + ( a + 1)^4 + 1 = 2 ( a^2 + (a + 1)^2 + 1) Again a^4 + ( a + 1)^4 + 1 = 2 ( a ^4 + 2 a^3 + 3 a ^2 + 2 a + 1) = 2 ( a^2 + a + 1)^2 This implies 2 ( a^2 + a + 1)^2 = 2 ( a^2 + (a + 1)^2 + 1) = 4 (a^2 + a + 1) Hereby a^2 + a + 1 = 2 ( a + 2)( a - 1)= 0. 2 ^ x = a = -2, 1

(16^x+(2^x+1)^4+1)/(4^x+(2^x+1)^2+1)=2 x=-Log[2,1+Sqrt[5]]=Log[((Sqrt[5]-1)/2),2]

Hoc est p = 3 solutio realis unus, respondam.

48 (1 /16^x + 1 / 32^x + 1/ 64^x) = 1/2^x - 1/16 ^ x 48 (4 ^ x + 2^x + 1) = 32 ^ x - 4 ^ x 48 ( 8 ^ x - 1) = 4 ^ x * ( 8 ^ x - 1) * ( 2 ^ x - 1) Now 8 ^ x - 4 ^x - 48 has a zero at x =2 or 2 ^ x = 4,

Sum is equal to 5 For 1minute question wasting 11 monutes is foolishness

माना कि 9p²-56=25 ( 5 का वर्ग) अर्थात (3p)²=25+56 = (3p)²= 81 = (3p)²=9² या (3×p)²=(3×3)² या(3×p)²/3=(3×3)²/3 या p²=3² अतः p=3✓ ans

p=3 satisfy.9p^2-56=81-56=25=5^2

9p^2-6p+1-57+6p=(3p+1)^2+6p-57 a perfect sq .if 6p=57=>p=19/3 but it is not integer but rational.

(3 p)^2 - q ^2 = 56 = 27 + 29 = 15^2 - 13^2 = 11 + 13 + 15 + 17 = 9^2 - 5^2 Hereby p = 3, -3, 5, -5

(9a^2)^2 ➖ (56)^2,={81a^4➖ 3136}= 2955a^4 .100^200^10^90^55a^4 10^10^10^20^10^9^10^5^11a^4 10^10^10^2^10^10^9^10^5^11^1a^4 2^5^2^5^2^5^2^2^5^2^5^9^2^5^5^1^1a^4 1^1^1^1^1^1^1^1^1^1^1^3^2^1^1^1 3^2a^2^2 3^1a^1^2 3a^2 (a ➖ 3a+2).

Too easy

x=2026

Sorry P is 5.

P=5. Then 9p^2= 225. So 225-56=169. Which is equal to 13^2. .So p = 13.

X=2026

P=+5 or -5

P=-5

2021/2022+2022/2023+2023/2024+2024/2025+2025/2021=5.0000024446871196771651136847981338696556510090406058894738699661958366764482729899217310685300550165931189484424686872182724695124443855963361798831087257849297587431823782940268476747022028458197746718762966764681369851606567086363141023161938626524168106643928410808643720271510558879496021884140267063879330865719705480646893952735734604224443243473712029508363099874547529646801149560735808152419934822841825785530350429982973365823183914811373227620581991895066400287009415743072983831464867774803719812397379720874645325695686636816892875527446420367650172317539792997550213825595471003380791518678988898120633556829832493197920173105852086665541947050725434156976542860922126942300143619720766285015393877436610875486258653738387902028225366806008847101547066162979049036068016495473514099058005581509914859141523946085006432411759334620762909096362120611832342964472839620552931174861492112423832078644463928432622370569867885893894384954785667748119669427046390180905456664852043189545873354564732614762393168642976713970340199997445786941230120878932595560378934962665616308381528971895220317146998928958434743019570664522594672792335040469052463344971295453107919389762153…

2021/2022+2022/2023+2023/2024+2024/2025+2025/2021>5 2021/2022+2022/2023+2023/2024+2024/2025+2025/2021=5 13805461121/5647128014820600

Greater than or equal to +or - 3

There are many sets of factors of a particular number. And if there is relation between sum of factors and anyone of factor please define as definition.

Please explain how you assume that multiplication product of factors of a number is equal to sum of factors or multiple of factors.

Sum is greater than 5.

3

(5^a*5^a*5^a)/(5^a+5^a+5^a)=5 a=Log[25,3]+0.5=(Log[5,3]+1)/2=Log[5,Sqrt[15]] final answer.

P=(±3,±5)

Problem:Let p is an integer such that 9p^2-56 is a perfect square. Find all values of P? 9P^2-56=J^2 J = ± 13, P = ± 5 J = ± 5, P = ± 3

p=19/2 as such 9(19/2)^2-56=)9.361-224)/4=3025/4=(55/2)^2.thus apart from being integer p=19/2 the expn. is perfect square.

81-56=25 is a perfect square, so my guess is p=±3.

Sum is multiple of 6,not factor of 6

{x+x ➖ }+{33+33 ➖ }/{35x+35x ➖ }{x+x ➖ }+{50+50 ➖ }/{29x+29x ➖}= {x+x ➖ }+{27+27 ➖ }/{64x+64x ➖ }={x^2+66}/70x^2+{x^2+100}/58x^2+{x^2+54}/128x^2={66x^2/70x^2+100x^2/58x^2+54x^2/128x^2}= 274x^6/240x^6 =1.34x^1 1.217x^1 1.2^17^1x^1 1.2^1^1x^1 2x^1 (x ➖ 2x+1).

Another method: 222 > 128 (=2⁷) Therefore, 222^333 > 128^333 i.e., 222^333 > (2⁷)^333 But (2⁷)^333=2^(7*333)=2^2331 Thus, we have 222^333 > 2^2331 -- (i) 333 < 512 (=2⁹) Therefore 333^222 < 512^222 i.e., 333^222 < (2⁹)^222 But (2⁹)^222=2^(9*222)=2^1998 Thus, 333^222 < 2^1998 -- (ii) From (i) and (ii) it may be concluded that 222^333 > 333^222

222^333=21654117651367595674009303657504323451466064914053983428477671597822265157425685169561874841163256524075141514993489271555967858593066483917829919872733053700297314467289961844505803918784072744460746150445290514796744806331743745976511664057316740779388759190888033837106991242331236052292951987109132788032026662242267174906775416166110573756353705185502865577802705607619967634905264590529587740625529408942008125192153154509343282558188666190135538294364099721627307609190367845633534496068314766553809195612358496978397339190752230161389547365660081596334681036837037703700666830359022006662309490025762988994974743847992366626595701202570543751286981134928210918727720874205524973256166024269504446167103612444566107103633363823845100063266133403756544212410020472363419697152

a>1,b>1,a <b natural ədəd olduqda 1.a+b<6 olarsa a^b <b^a 2.a+b=6 olarsa a^b=b^a 3.a+b>6 olarsa a^b>b^a. Təşəkkürlər.

x=37

It’s in my head.

k*k*k-k*k=100 k=5 k=-2±4i

333^222=96077089512754103040990584837623202364496756594795428342777025553966634920121404896310535002071538497201655056712164231099293209315294327763722277282685173052612925303420665872433505353462948853028311578606415210856013749109602950641879226996769232529415724120539234746388718319329476626940519292142029465642709934617779087779795455010273843755877782516581901176301148742756378477198298069591132115948171777872762718463682113867052684983634890740215626792287710181292675895338947327665435892857350703927655360721930144668771272062044490454077589846687626523289 222^333=21654117651367595674009303657504323451466064914053983428477671597822265157425685169561874841163256524075141514993489271555967858593066483917829919872733053700297314467289961844505803918784072744460746150445290514796744806331743745976511664057316740779388759190888033837106991242331236052292951987109132788032026662242267174906775416166110573756353705185502865577802705607619967634905264590529587740625529408942008125192153154509343282558188666190135538294364099721627307609190367845633534496068314766553809195612358496978397339190752230161389547365660081596334681036837037703700666830359022006662309490025762988994974743847992366626595701202570543751286981134928210918727720874205524973256166024269504446167103612444566107103633363823845100063266133403756544212410020472363419697152

222^333=21654117651367595674009303657504323451466064914053983428477671597822265157425685169561874841163256524075141514993489271555967858593066483917829919872733053700297314467289961844505803918784072744460746150445290514796744806331743745976511664057316740779388759190888033837106991242331236052292951987109132788032026662242267174906775416166110573756353705185502865577802705607619967634905264590529587740625529408942008125192153154509343282558188666190135538294364099721627307609190367845633534496068314766553809195612358496978397339190752230161389547365660081596334681036837037703700666830359022006662309490025762988994974743847992366626595701202570543751286981134928210918727720874205524973256166024269504446167103612444566107103633363823845100063266133403756544212410020472363419697152