ABC is a triangle in which \(R\) represents the circumradius and \(r\) represents the inradius and \(d\)represents the distance between circumcentre and incentre . Then find \(d\) in terms of \(R,r\)

ABC is a triangle . Point P is on side BC . \(r_{1},r_{2},r\) are inradii of Triangle ABP,ACP,ABC respectively. \(h_{a}\) is the altitude of ABC from A . Prove that \(\frac{1}{r_{3}}+\frac{1}{r_{2}}-\frac{r}{r_{2}r_{3}}=\frac{2}{h_{a}}\)

If \(a,b,c\)are real numbers such that \(a^{2}=bc , a+b+c=abc\), then prove that \(a^{2}\geq3\).

If \((x,y,z)\) are prime numbers then find all solutions of \(x(x+y)=120+z\).

5.Let ABC be an acute angled triangle with BC > AC. Let O be the circumcentre , H be the orthocentre of the triangle ABC . CF is the altitude . The perpendicular to OF at E meets the side CA at P . Prove that angle FHP = angle BAE.

6.Is it possible to write the numbers \(1,2,3....,121\) in an \(11 \times 11\) table so that any two consecutive numbers be written in cells with a common side and all perfect squares lie in a straight column?

7.The positive integers are seprated into two subsets with no common elements. Show that one of these subsets must contain a three term A.P.

Prove that : If three lines from vertices of a triangle are concurrent then their isogonals are also concurrent

\(ω=\sum_{n=1}^{2014}\sqrt{1+\frac{1}{n^{2}}+\frac{1}{(n+1)^{2}}}\). Find \(ω\).

Please reshare this note so that more people get to know about it. This note contain all the questions asked in the exam . Enjoy solving these questions. Remember the guidelines \(\color{red}{''novel..and...elegant...solutions.. will ..get ..extra ...credits.''}\).

## Comments

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TopNewest3) \(a^3=abc=a+b+c\ge a+2\sqrt{bc}=3a\implies a^2\ge 3\)

4) If \(x\) odd, \(y\) even, then \(y=2\implies x(x+2)=120+z\)\(\implies x^2-2x-120=z\)\(\implies (x-12)(x+10)=z\)\(\implies x=13, z=23\)

Else \(x(x+y)\) is even so \(z=2\) and \(x(x+y)=122=2\times 61\implies x=2, y=59\)

Only solutions are \((x,y,z)=(13, 2, 23), (2, 59, 2)\)

6) No. Just write the squares \(1^2, 2^2, \ldots , 11^2\) in a column and start filling in squares from \(1\). You will find that all moves are forced, and will not create a \(11\times 11\) shape. – Daniel Liu · 1 year, 4 months ago

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– Surekha Jadhav · 1 year, 4 months ago

Daniel liu can you explain your answer to problem 6 ?Log in to reply

WLOG let the numbers \(1^2, \ldots 11^2\) be going from top to bottom. Consider the stretch of numbers from \(k^2\) to \((k+1)^2=k^2+2k+1\). There are \(2k\) numbers from these two, which gives \(k\) numbers away and \(k\) numbers back as the farthest distance this sequence of numbers can go. However, that means that no matter what \(k\) we choose, we cannot reach either the top left or top right square, depending on where the column is. Thus, that square cannot be labelled by a square, and we are done. – Daniel Liu · 1 year, 4 months ago

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– Xuming Liang · 1 year, 4 months ago

I don't think the "WLOG" part is legal, it could be possible for \(1^2\) to start in the middle and swiggle its way through. I posted a solution for it.Log in to reply

– Daniel Liu · 1 year, 4 months ago

It's given that \(1^2, 2^2, \ldots , 11^2\) all lie on a column. I'm just saying to order them from top to bottom without loss of generality.Log in to reply

– Shivam Jadhav · 1 year, 4 months ago

Please help in solving problem number 2.Log in to reply

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– Shivam Jadhav · 1 year, 4 months ago

Can you solve the second question?Log in to reply

– Neel Khare · 4 months, 3 weeks ago

what is r3 in the second questionLog in to reply

– Shivam Jadhav · 4 months, 2 weeks ago

r3=r1Log in to reply

Q 4)

Case 1:If \(z\) is an even prime i.e. \(z=2\). Then \(x(x+y) = 2 \times 61\). But since \(x+y > x\), the only possibility is \(x=2\) and \(y=59\). So, one solution in this case \((2,59,2)\).Case 2:If \(z\) is an odd prime. Then \(x(x+y)\) is odd, which implies that both \(x\) and \(x+y\) giving out that \(y\) is even. But \(y\) is a prime, so \(y=2\). So the equation transforms into \(x^2 + 2x -(120+z)=0\). So, the discriminant should be a perfect square.\[\implies \Delta = 4z + 484 =k^2 \quad ; \quad k \in \mathbb{Z}\] \[\implies 2|k\]

So, letting \(k=2l\).

\[\implies z = (l-11)(l+11)\]

But \(z\) is a prime. So, \(l-11=1\) and \(l+11=z\), which gives us that \(z=23\). Substituting in the above equation gives out that \(x=11\). Therefore \((11,2,23)\) is only solution in this case.

Therefore, total number of solutions are \(2\). And they are \((11,2,23)\) and \((2,59,2)\). – Surya Prakash · 1 year, 4 months ago

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One question. Is Ramanujan the senior round? – Swapnil Das · 5 months ago

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@Shivam Jadhav here is the proof for Q1 – Sualeh Asif · 1 year, 4 months ago

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– Shivam Jadhav · 1 year, 4 months ago

Please try second questionLog in to reply

@Daniel Liu @Surya Prakash @Calvin Lin @Best of Geometry @Xuming Liang please help me in solving question number 2. – Shivam Jadhav · 1 year, 4 months ago

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Q 7) 2) https://brilliant.org/problems/simply-awsome/ Similar problem – Surya Prakash · 1 year, 4 months ago

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1)d^2=R^2-2

Rr – Aakash Khandelwal · 1 year, 4 months agoLog in to reply

The answer to Q.4 is (2,59,2) and (11,2,23). So there are two solutions. – Kushagra Sahni · 1 year, 4 months ago

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I had done Q1 about 1 year ago. The relation is \(d=\sqrt{R(R-2r)}\). It is famous and called as Euler Triangle formula. Its proof is based on cyclicity and a bit of trigonometry. – Nihar Mahajan · 1 year, 4 months ago

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– Shivam Jadhav · 1 year, 4 months ago

You need to prove it.Log in to reply

– Nihar Mahajan · 1 year, 4 months ago

I have proved it.Log in to reply

– Shivam Jadhav · 1 year, 4 months ago

Where?Log in to reply

@Nihar Mahajan @Surya Prakash – Shivam Jadhav · 1 year, 4 months ago

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@Calvin Lin

@niharmahajan – Shivam Jadhav · 1 year, 4 months ago

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