Maths RMO Problems(1)........

Hey friends.

I mean Brilliantians I am back with some amazing problems which are generally asked in the RMO-INMO level examination .I am sharing the image of the paper containing the questions.

Please try and if possible send the solutions

.Also it would be great if you all participate in sharing the questions from your own. I would also be sharing problems based on NSEP level. Thanks

Dont forget to share and like this.............

Note by Abhisek Mohanty
3 years, 2 months ago

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Q13 The sum of the digits of any number formed using the given conditions is 1 + 4 + 9+ . . . . + 81 = 285 = 3(95) which implies the number is divisible by 3 but not by 3 squared which is 9. Therefore any number formed using the given conditions is not a perfect square

Alan Joel - 3 years ago

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question 11 given 34x=43y =>34x+43x=43(y+x) =>77x=43(x+y) now 43 does not divide 77 hence x+y contains 77 i.e-11*7 hence x+y is not prime.

ALEKHYA CHINA - 3 years, 2 months ago

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That does not exclude the possibility of x+y being odd but not prime

Abdur Rehman Zahid - 3 years, 2 months ago

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Can you please explain me what u are trying to say?

ALEKHYA CHINA - 3 years, 2 months ago

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Which grade you in ?

Rajdeep Dhingra - 3 years, 2 months ago

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Question 9 We shall analyze 2 cases. Case 1 Either of a,ba,b is even.

WLOG let aa be even.Then ab(ab)=45045ab(a-b)=45045 is even.But 45045 is odd,contradiction. Hence no solutions exist in this case.

Case 2 Both a,ba,b are odd.

Then aba-b is even.Therefore ab(ab)=45045ab(a-b)=45045 is even,contradiction.

Hence no solutions exist.

Abdur Rehman Zahid - 3 years, 2 months ago

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Question 8(i)

Let the roots be a,b,c,da,b,c,d.(Note that the roots are positive).Then: p=(a+b+c+d)q=ab+ac+ad+bc+bd+cdr=(abc+abd+acd+bcd)s=abcd\begin{aligned} p&=-(a+b+c+d)\\ q&=ab+ac+ad+bc+bd+cd\\ r&=-(abc+abd+acd+bcd)\\ s&=abcd \end{aligned} pr16s0    (a+b+c+d)(abc+abd+acd+bcd)16abcdpr-16s\geq 0\implies (a+b+c+d)(abc+abd+acd+bcd)\geq 16abcd which follows by applying AM-GM on each term.

I couldn't understand;what does the variable "a" denote in Q 8(ii)?

Abdur Rehman Zahid - 3 years, 2 months ago

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Question 1 x0,1,2,3,4,5,6(mod7)x30,1,1(mod7)\begin{aligned} x &\equiv 0,1,2,3,4,5,6 \pmod{7}\\ x^3 &\equiv 0,1,-1\pmod{7} \end{aligned}

Assume that either of a,b,ca,b,c is a multiple of 7.

Then obviously abc(a3b3)(b3c3)(c3a3)abc(a^3-b^3)(b^3-c^3)(c^3-a^3) is a multiple of 7.

So now WLOG assume that a,b,c≢0(mod7)a,b,c\not\equiv 0\pmod{7}.Then a3,b3,c31,1(mod7)a^3,b^3,c^3\equiv 1,-1\pmod{7}.There are 2×2×2=82\times 2\times 2=8 different possible cases corresponding to the different values of a3,b3a^3,b^3 and c3c^3 modulo 7,which are: Values modulo 7a3b3c3Case 1111Case 2111Case 3111Case 4111Case 5111Case 6111Case 7111Case 8111\begin{array}{c|c|c|c} \text{Values modulo 7} & a^3 & b^3 & c^3 \\ \hline \text{Case 1} & 1 & 1 & 1 \\ \hline \text{Case 2} & -1 & -1 & -1 \\ \hline \text{Case 3} & 1 & -1 & -1 \\ \hline \text{Case 4} & -1 & 1 & -1 \\ \hline \text{Case 5} & -1 & -1 & 1 \\ \hline \text{Case 6} & -1 & 1 & 1 \\ \hline \text{Case 7} & 1 & -1 & 1 \\ \hline \text{Case 8} & 1 & 1 & -1 \end{array}

Observe that,because of the symmetry of the expression,Cases 3,4,5 and Cases 6,7,8 are equivalent.Therefore,we only need to check Case 1,2,3 and 6.Simply evaluate the cases to get that abc(a3b3)(b3c3)(c3a3)0(mod7)    a,b,cZabc(a^3-b^3)(b^3-c^3)(c^3-a^3)\equiv 0\pmod{7}\;\forall \;a,b,c\in \mathbb{Z}

Abdur Rehman Zahid - 3 years, 2 months ago

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Nice solution bro..........upvoted

Abhisek Mohanty - 3 years, 2 months ago

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vmc questions

Piyush Kumar Behera - 3 years, 2 months ago

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Question 2

(xy7)2=x2+y2x2y214xy+49=x2+y2x2y212xy+36+13=x2+y2+2xy(x+y+xy6)(x+yxy+6)=13=13×1=1×13=13×1=1×13(x,y)=(3,4),(4,3),(0,7),(7,0)(xy-7)^2=x^2+y^2\Rightarrow x^2y^2-14xy+49=x^2+y^2 \\ x^2y^2-12xy+36+13=x^2+y^2+2xy \\ (x+y+xy-6)(x+y-xy+6)=13=13×1=1×13=-13×-1=-1×-13 \\ (x,y)=(3,4),(4,3),(0,7),(7,0)

Akshat Sharda - 3 years, 2 months ago

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Nice solution................upvoted....

Abhisek Mohanty - 3 years, 2 months ago

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Can anyone recommend me some good books for INMO and and other maths olympiad??????????

Abhisek Mohanty - 3 years, 2 months ago

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