×

# RMO Practice Board!

Post as many problems as you can for others to try.

Please don't post problems from past year RMO papers.

Have FUN!

Let $$a, b, c, d, e, f$$ be real numbers such that the polynomial equation

$x^{8} - 4x^{7} +7x^{6} +ax^{5}+ bx^{4} + cx^{3} + dx^{2} + ex + f = 0$

has eight positive real roots. Determine all possible values of $$f$$

Note by Harsh Shrivastava
1 year, 8 months ago

Sort by:

Since we were given that the roots were positive reals,we get motivated and tempted to use,$$R.M.S\geq A.M\geq G.M\geq H.M$$,also since we have been given,$$\sum \alpha$$ and $$\sum \alpha\beta$$ we try to find the value of $$\sum \alpha^2=4^2-2\times 7=2$$.We apply $$R.M.S \geq A.M$$ $\sqrt{\dfrac{2}{8}} \geq \dfrac{4}{8}\\ \Longrightarrow \dfrac{1}{2} \geq \dfrac{1}{2}$,hence equality occurs,meaning that all the roots are equal.Now,that means,$$A.M=G.M$$,hence$\dfrac{4}{8}=\sqrt[8]{f}\\ \Longrightarrow \dfrac{1}{256}=f$,hence there is only one value of $$f=\dfrac{1}{256}$$.Is this correct Harsh? · 1 year, 8 months ago

In your solution you need prove that $(\sum_{cyclic}ω)^{2}-\sum_{cyclic}ω_{1}ω_{2}=\sum_{cyclic}ω^{2}$ · 1 year, 8 months ago

That's an identity that is true for all numbers, real and imaginary. And there should be a 2 infront of that second summation. · 1 year, 6 months ago

Yes , its one of the short Newton identities. · 1 year, 6 months ago

If $$x,y,z$$ are positive real numbers, prove that,

$\large (x+y+z)^2 (yz+zx+xy)^2 \le 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$

Give me as much different solutions please . · 1 year, 8 months ago

U may use the concept of fermat pt to solve this one..then it will be very easy...suppose three line segments AF,BF,CF of length x,y,z meet each other at120 degrees and now use cosine rule to find out the lengths of the triangle and see that each term on the rhs of the inequality is actually squares of the sides...then express xy+yz+zx as $$\frac{4}{\sqrt{3}}T$$ where T is the area of the triangle....then use $$T=\frac{abc}{4R}$$ and see that the inequality reduces to 3R greater than or equal to (x+y+z) which is obvious since F is the fermat pt · 1 year, 7 months ago

You have a set of consecutive positive integers,$$\{1,2,3,...,199,200\}$$,if you choose $$101$$ numbers from these at random,then prove that there at-least two numbers of which one divides the other. · 1 year, 7 months ago

Find all positive integer solutions $$a$$, $$b$$ and $$c$$ such that

$\dfrac {7}{a^2} + \dfrac {11}{b^2} = \dfrac {c}{77}.$ · 1 year, 7 months ago

$$\text{THIS IS ONLY AN OUTLINE OF THE SOLUTION, NOT A COMPLETE SOLUTION.}$$

$\dfrac {7 }{a^2} + \dfrac {11 }{b^2} = \dfrac{c}{77}.$

Case 1: $$c>1$$

This forces $$a^{2} < 7 \text{ and } b^{2} < 11$$

Now it is easy to check that $$(a,b,c) = (1,1,1386) \text{ and } (a,b,c) = (3,3,154)$$ satisfy the given conditions.

Case 2: $$c \leq 77$$

Since $$c \leq 77$$ this implies $$\dfrac{c}{77} \leq 1$$.

Let $$\dfrac{c}{77} = \dfrac{1}{k}$$ for some integer $$k \geq 1$$.

$$\implies c = \dfrac{77}{k}$$

Now since $$c$$ is a positive integer , this forces $$k \text{ | } 77$$.

Possible values for $$k = 1,7,11,77$$.

Now plugging the possible values of $$c$$ in the given equation and using Simon's Favourite Factoring Trick, we can easily solve for $$a,b$$ · 1 year, 7 months ago

If $$x$$ and $$y$$ are integers such that $$y^2+3(xy)^2=30x^2+517$$, Prove that $$3(xy)^2=588$$ · 1 year, 8 months ago

Let $$y^{2} = b$$ and $$x^{2} = a$$.

$$b(1 + 3a) = 30a +517$$

$$\implies b(1+3a) -10(3a + 1) = 507$$

$$\implies (b-10)(3a+1) = 507 \times 1 = 13 \times 39 = 3 \times 169$$

Since a , b are integers, after checking through all 6 cases, we get only admissible values of $$(x ,y) = ( \pm 2, \pm 7)$$.

This forces $$3(xy)^{2}= 588$$.

$$\QED$$ · 1 year, 8 months ago

Let $$x,y$$ and $$z$$ be positive real numbers such that $$x+y+z=1$$. Prove that $$\frac{yz}{1+x}+\frac{zx}{1+y}+\frac{xy}{1+z} \le \frac{1}{4}$$. · 1 year, 7 months ago

I got this. This is simple AM-HM inequality. Write 1+x=(1-y)+(1-z). Similarly for others

Apply AM-HM and then manipulate on the RHS. · 1 year, 7 months ago

x, y, z are positive real numbers such that x^2+y^2+z^2=3 Prove that x+y+z >= (xy)^2+(yz)^2+(zx)^2

Sorry for not using latex as I was in a hurry. · 1 year, 7 months ago

Can u send the solution of this question ? I have been trying. No progress .... · 1 year, 7 months ago

I didn't get it. · 1 year, 7 months ago

Find all primes $$p$$ such that for all prime $$q$$, the remainder of $$p$$ upon divison by $$q$$ is squarefree. · 1 year, 8 months ago

I think that 3 is the only prime number. I used the fact that all primes>3 is of the form $$\pm 1 \pmod{6}$$. · 1 year, 7 months ago

In a triangle $$ABC$$ , D is the midpoint of side AB and E is the point of trisection of side BC nearer to C.

Given that $$\angle ADC = \angle BAE$$, find $$\angle BAC$$. · 1 year, 7 months ago

Is the answer angle BAC = 90 degrees ? I hope so My proof : Apply Menelaus theorem on triangle BCD with transversal EA. Obtain that F is the midpoint of CD. Then its simple angle chasing :) · 1 year, 7 months ago

Comment deleted Dec 04, 2015

Any solution is welcome. How did u solve using Trigonometry ? · 1 year, 7 months ago

Let $$\angle FDA = \angle FAD = \alpha \ , \ \angle AEC=\gamma$$.Let $$FD=FA=x \ , \ CE=y \ , \ BE=2y$$. Using Sine Rule in $$\Delta ADF$$ , we get $$AD=2x\cos(\alpha)$$ and thus $$AB=4x\cos(\alpha)$$. By angle chase we have $$\angle CFE= 180-2\alpha$$ and again applying Sine Rule to $$\Delta CEF$$ , we get $$FC=\dfrac{y\sin(\gamma)}{sin(2\alpha)}$$. Now apply Sine Rule to $$\Delta ABE$$ and knowing that $$\sin(180^\circ - \gamma)= \sin(\gamma)$$ , we get $$x=\dfrac{y\sin(\gamma)}{sin(2\alpha)}= FC$$ . This means $$F$$ is midpoint of $$DC$$ , and we get the desired result. · 1 year, 7 months ago

Nice work there ! · 1 year, 7 months ago

Yes , Menelaus is indeed a good approach for this problem. For shortening the solution: Once you get F is the midpoint of CD , you can directly say that $$\angle BAC = 90^\circ$$ , do you see why? · 1 year, 7 months ago

FD=FC=FA . Great ! · 1 year, 7 months ago

Do u have some problems for practice ? I need some number theory. I am not that good or in other words I am bad in it · 1 year, 7 months ago

I also need some NT probs... · 1 year, 7 months ago

Even I need NT problems... · 1 year, 7 months ago

Are you going to appear for RMO from mumbai region or from the pune region ? · 1 year, 7 months ago

Pune region. · 1 year, 7 months ago

Yup I also got 90 degrees. · 1 year, 7 months ago

Shrihari,we get that $$CF=2DF$$ · 1 year, 7 months ago

I don't think so check again. · 1 year, 7 months ago

Applying Menelaus' theorem, ( (BE/EC)(CF/FD)(DA/AB)=1 · 1 year, 7 months ago

BE/EC is 2 and DA/AB is 1/2. So CF/FD is 1 · 1 year, 7 months ago

Let $$p>3$$ be a prime number. Suppose $$\displaystyle \sum_{k=1}^{p-1}{\frac{1}{k}}=\frac{a}{b}$$ where $$\gcd(a,b)=1$$. Prove that $$a$$ is divisible by $$p^{2}$$. · 1 year, 7 months ago

I am able to prove that a is divisible by p. But not able to prove divisible by p^2. Could u send the solution ? · 1 year, 7 months ago

Please post the proof, thanks! · 1 year, 7 months ago

My proof is incomplete I could not prove that a is divisible by p^2 · 1 year, 7 months ago

Post the proof that a is divisible by p, Thanks! · 1 year, 7 months ago

Take LCM on the LHS and try proving that each term in the numerator leaves a different residue modulo p. Do it by contradiction. Once you are able to prove that, the rest is clear. If it is difficult, first take a smaller example say p=7. Then work it out · 1 year, 7 months ago

I have the solution only till $$a$$ is divisible by $$p$$. To prove that $$a$$ is divisible by $$p^{2}$$ is given below the solution as a challenge. · 1 year, 7 months ago

If $$a,b \ge 0$$. Prove that $$\sqrt{2}(\sqrt{a(a+b)^{3}}+b \sqrt{a^{2}+b^{2}}) \le 3(a^{2}+b^{2})$$. · 1 year, 7 months ago

Let $$a,b,c$$ be positive reals satisfying $$(a+b)(b+c)(c+a) = 1$$.

Prove that $$ab+bc+ca \leq \frac{3}{4}$$ · 1 year, 7 months ago

Using am gm

(a + b) + (b + c) + (c + a) > 3

a + b + c > 3/2

now using cauchy,

$$3(a^2 + b^2 + c^2) > (a + b + c)^2$$

so

$$a^2 + b^2 + c^2 > 3/4$$

using cauchy again,

$$(a^2 + b^2 + c^2)^2 > (ab + bc + ca)^2$$

3/4 > ab + bc + ca · 1 year, 7 months ago

Dev,your solution is incorrect. Check the solution again,you will find your flaw. · 1 year, 7 months ago

Nice solution. You could also have used the fact that $$(a+b+c)^{2} \ge 3(ab+bc+ca)$$ · 1 year, 7 months ago

You got the following two inequalities,$a^2+b^2+c^2>ab+bc+ca\\ and\\ a^2+b^2+c^2>\dfrac{3}{3}$,what next?How can you conclude that,$\dfrac{3}{4}>ab+bc+ca$? · 1 year, 7 months ago

using cauchy again, like this

(sum(a^2))(sum(b^2)) > (sum(ab))^2

sum is cyclic · 1 year, 7 months ago

Let $$a,b,c,d \in \mathbb{N}$$ such that $$a \ge b \ge c \ge d$$ and $$d \neq 0$$. Show that the equation $$x^4 - ax^3 - bx^2 - cx -d = 0$$ has no integer solution in $$x$$.

Please post a "complete solution" like you would do in RMO. · 1 year, 7 months ago

I'll have a crack at it. As $$d \neq 0$$, $$x \neq 0$$.

Denote $$y = \frac{1}{x}$$.

We have

$x^4 = ax^3 + bx^2 + cx + d \quad \quad$

$1 = ay + by^2 + cy^3 + dy^4 \quad \quad (1)$

Now, we separate this into 2 cases:

Case 1:

$$x = -n$$ where $$n \in \mathbb{N}$$, then $$y=-\dfrac{1}{n}$$. Substituting this into $$(1)$$, we get

$\dfrac {-a}{n} + \dfrac {b}{n^2} + \dfrac {-c}{n^3} + \dfrac {d}{n^4} = 1$

$\dfrac {-an+b}{n^2}+\dfrac{-cn+d}{n^4}=1$

Here, the LHS is non-positive since $$a \geq b$$ and $$c \geq d$$, whereas the RHS is positive. Thus no solutions exist in this case.

Case 2:

$$x = n$$ where $$n \in \mathbb{N}$$, then $$y=\dfrac{1}{n}$$. Substituting this into $$(1)$$, we get

$\dfrac {a}{n} + \dfrac {b}{n^2} + \dfrac {c}{n^3} + \dfrac {d}{n^4} = 1$

This implies that $$n \neq 1$$, and

$\dfrac {a}{n} < 1$

$1 \leq \dfrac {a}{n} + \dfrac {a}{n^2} + \dfrac {a}{n^3} + \dfrac {a}{n^4} < \displaystyle \sum_{i=1}^{\infty} \dfrac{a}{n^i} = \dfrac {a}{n-1}$

These two equations imply that

$\dfrac {n}{a} > 1$

$\dfrac {n-1}{a} < 1$

There are no such $$n$$ that satisfy the above two inequalities. Thus, no solutions exist. · 1 year, 7 months ago

Isn't this from INMO 2013 ?? · 1 year, 6 months ago

Yes it is. · 1 year, 6 months ago

Hey why aren't u participating in the INMO practice board ? · 1 year, 6 months ago

Let $$p$$ and $$q$$ be distinct primes. Show that

$$\displaystyle \sum_{j=1}^{(p-1)/2}[\frac{qj}{p}]+ \sum_{j=1}^{(q-1)/2}[\frac{pj}{q}]=\dfrac{(p-1)(q-1)}{4}$$.

Note that $$[ \ ]$$ represents the greatest integer function. · 1 year, 7 months ago

Let $$S={(x,y)|x,y \in N, \quad 1 \le x \le (p-1)/2, \quad 1 \le y \le (q-1)/2}$$.

We note that the set $$S$$ can be partitioned into two sets $$S_1$$ and $$S_2$$ according as $$qx>py$$ or $$qx<py$$. Note that there are no pairs in such $$S$$ such that $$qx=py$$.

The set $$S_1$$ can be described as the set of all pairs $$(x,y)$$ satisfying $$1 \le x \le (p-1)/2, \quad 1 \le y < qx/p$$. Then $$S_1$$ has $$\displaystyle \sum_{j=1}^{(p-1)/2}{\frac{qj}{p}}$$ elements.

Similarly, $$S_2$$ has $$\displaystyle \sum_{i=1}^{(q-1)/2}{\frac{pi}{q}}$$. Hence, we get the required result. · 1 year, 7 months ago

Find all primes $$x,y$$ such that $$x^{y}-y^{x}=xy^{2}-19$$.Enjoy solving this. · 1 year, 7 months ago

Using FLT, we have

$x^y \equiv x \pmod{y} \Rightarrow x \equiv -19 \pmod {y} \Rightarrow x+19 \equiv 0 \pmod{y}$

$y^x \equiv y \pmod{x} \Rightarrow -y \equiv -19 \pmod {x} \Rightarrow 19-y \equiv 0 \pmod{x}$

From here, it is clear that either $$x=2$$ or $$y=2$$. We now separate this into 2 cases:

Case 1: $$y=2$$

When $$y=2$$, we have $$19 - 2 = 17 \equiv 0 \pmod{x}$$, which implies $$x=17$$. The solution in this case is $$(17, 2)$$. Checking, however, we find that this clearly doesn't work. So, there are no solutions in this case.

Case 2: $$x=2$$

When $$x=2$$, we have $$19+2 = 21 \equiv 0 \pmod {y}$$, which implies $$y=3$$ or $$y=7$$. Checking, we find both these solutions work. Thus, 2 solutions exist in this case: $$(2, 3)$$ and $$(2,7)$$.

Therefore, only 2 solutions exist: $$(2, 3)$$ and $$(2,7)$$. · 1 year, 7 months ago

would you please explain how x = -19 mody · 1 year, 7 months ago

Putting $$x^y-y^x=xy^2-19$$ in $$\pmod{y}$$, we get $$x=-19 \pmod{y}$$. · 1 year, 7 months ago

Using FLT and modulo $$x$$ and $$y$$ we get that,$$x+19 \equiv 0 \pmod{y},y-19 \equiv 0 \pmod{x}$$,now just use parity(as suggested by @Sharky Kesa). · 1 year, 7 months ago

If $$a,b,c$$ are positive integers such that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}<1$$. Prove that the maximum value of $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$$ is $$\dfrac{41}{42}$$ · 1 year, 8 months ago

If none of $$a,b,c=2$$, the maximum value of the expression will be $$\frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{47}{60}<\frac{41}{42}$$. So, one of them has to $$2$$ say $$a$$. So we have to prove that the maximum value of $$\frac{1}{b}+\frac{1}{c}$$ is $$\frac{10}{21}$$. If none of $$b,c=3$$, the maximum of the expression will be $$\frac{1}{4}+\frac{1}{5}=\frac{9}{20}<\frac{10}{21}$$. So, one them has to be $$3$$ say $$b$$. Therefore, $$\frac{1}{c} \le \frac{1}{7}$$.Therefore the maximum value of $$c$$ is $$7$$ and the maximum value of $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ is $$\frac{41}{42}$$.

Not sure if this is the best method. · 1 year, 7 months ago

Find all reals $$x, y, z$$ in (1,∞) such that

$x + y + z + \dfrac{3}{x-1}+ \dfrac{3}{y-1} + \dfrac{3}{z-1} = 2(\sqrt{x+2} + \sqrt{y+ 2} + \sqrt{z+2})$ · 1 year, 8 months ago

https://brilliant.org/problems/three-variables-one-equation/?group=4bQcwX9pLM2R#!/solution-comments/106741/ · 1 year, 7 months ago

Let $$x,y$$ be real numbers such that $$x,y > 1$$

Prove that $$\dfrac{x^{2}}{y-1} + \dfrac{y^{2}}{ x-1} \geq 8$$ · 1 year, 7 months ago

Just use AM-GM inequality directly on the expression in the LHS and then use the fact that (a-2)^2>0 Then rearrange a bit. You get the required answer · 1 year, 7 months ago

Alternatively we can substitute p = x-1 and q = y-1.

Then it will be easy manipulate the expression. · 1 year, 7 months ago

Nice idea! · 1 year, 7 months ago

Since the expression is symmetric, let's assume without loss of generality that $$x \ge y$$.

$$\frac{x^{2}}{y-1}+\frac{y^{2}}{x-1} \ge \frac{y^{2}}{y-1}+\frac{y^{2}}{x-1} \ge 4+\frac{y^{2}}{x-1} \implies \frac{x^{2}}{y-1} \ge 4$$. Minimum value of $$\frac{x^{2}}{y-1}$$ is $$4$$ and it occurs if and only if $$x=y=2$$.

Therefore, $$\frac{x^{2}}{y-1}+\frac{y^{2}}{x-1} \ge 8$$. · 1 year, 7 months ago

Hmm you have not proven that min value of $$\frac{x^{2}}{y-1}$$ is 4. · 1 year, 7 months ago

Do you mean that I haven't proven that minimum value of $$\frac{y^{2}}{y-1}$$ is $$4$$? · 1 year, 7 months ago

Everyone here who have posted questions on this board and have not got any solutions please post the solutions now as RMO is only 42 hours from now. Its better that we understand the methods. Sorry that I cannot send the solution of my geometry question as I myself have been trying that question but could not arrive at the solution.

And please post some more questions if possible :) · 1 year, 7 months ago

Done :) · 1 year, 7 months ago

Thanks · 1 year, 7 months ago

$$1991^{k}|(1990^{1991^{1992}}+1992^{1991^{1990}})$$

$$max(k)=?$$

Details:

$$b|a\rightarrow$$ $$a$$ is divisible by $$b$$.

· 1 year, 7 months ago

1991 · 1 year, 7 months ago

Yes..... Can you post solution ? · 1 year, 7 months ago

How ? Could u give some rough outline of your solution ? · 1 year, 7 months ago

Shrihari,I have posted the general result of that combinatorics problem in its 'discuss solution' part. Now try to prove the general result .

https://brilliant.org/problems/triangular-grid-problem/?group=U3Rx70QSpE0z&ref_id=1044906 · 1 year, 7 months ago

Hey Saarthak I haven't heard of Catalan numbers before. You need to elaborate a bit Could you give a proof for the number of ways ? It will help me understand better. Thanks in advance · 1 year, 7 months ago

Try this set guys. · 1 year, 7 months ago

Here is a geometry question. Waiting for numerous solutions. In a triangle ABC, excircle opposite to A touches BC at point Q. Let M be the midpoint of BC. Incircle of triangle ABC touches BC at point P. Prove that PM=MQ. I have been trying this question but I am stuck. So please solve this question. · 1 year, 7 months ago

Its easy , note that $$BP=QC=s-b$$ where $$s$$ is the semiperimeter of the triangle and $$b$$ is length of side opposite to $$\angle B$$ respectively. Since $$M$$ is mid point of $$BC$$ , the result follows. · 1 year, 7 months ago

Oh nice ! · 1 year, 7 months ago

Are protractors allowed for RMO? Nothing about it is mentioned in the website and the hall ticket. · 1 year, 7 months ago

In the instructions given in question paper , it is said that protractors are not allowed. · 1 year, 7 months ago

Let d(k) denote the sum of the digits of k in base10. Find all natural numbers n such that n, d(n), d(d(n)) sum to 2015. · 1 year, 7 months ago

I am getting no solution. Is that right ? My proof : n,d(n),d(d(n)) are congruent to the same number say 'x' (mod 9). So their sum is congruent to 3x (mod 9). Therefore 2015 is congruent to 3x ( mod 9) which is impossible as 3 does not divide 2015 · 1 year, 7 months ago

Find all pairs $$(m,n)$$ such that $$2^m+3^n$$ is a perfect square. (A popular NT problem) · 1 year, 7 months ago

Its easy.Only one solution (2,2) · 1 year, 7 months ago

Wrong. Check again. · 1 year, 7 months ago

I think he means (4,2). · 1 year, 7 months ago

Ya I meant (4,2). Sorry,typing error. :P · 1 year, 7 months ago

Its okay , please post your method. :) · 1 year, 7 months ago

Taking mod 4 we get that m,n should be even. Then using $$gcd({2}^{m},3^{n})=1$$, we use the general formula for pythagorean triplets. Taking various cases,we get that only $$m=4,n=2$$ are the solutions. · 1 year, 7 months ago

If $$m$$ is a positive real number and satisfies the equation $2m=1+\frac{1}{m}+\sqrt{m^{2}-\frac{1}{4}}$ Find $$m$$. · 1 year, 7 months ago

Could you send the solution of this question. I tried expanding and landed up with a fourth degree polynomial whose roots I am unable to guess · 1 year, 7 months ago

Prove that $2 < (\dfrac{n+2}{n+1}) ^ {(n+1)}$ for all integers $$n > 0$$. · 1 year, 7 months ago

Well its easy just use Bernoulli's inequality that (1+x)^n > 1+nx · 1 year, 7 months ago

Prove that

$$1<\frac { 1 }{ 1001 } +\frac { 1 }{ 1002 } +.........+\frac { 1 }{ 3001 } <\frac { 4 }{ 3 }$$ · 1 year, 7 months ago

It is easy. I got a calculus and a non calculus answer · 1 year, 7 months ago

Since Rmo is a precalculus olympiad, answer without calculus will be better:P · 1 year, 7 months ago

Can u tell me how to add an image in the answer? · 1 year, 7 months ago

Alternate text

. Type the text which you wish to be displayed if the image doesn't load in the third brackets, and paste the image URL in the second brackets. In the following example, the image URL is http://s12.postimg.org/5qhlgca71/untitled.png, and I wrote the code

This is an image file :D

. Here's how it appears:

this is a copy paste from this note on moderators · 1 year, 7 months ago

I think we can use calculus in rmo .But if we use it we need to be very careful in our steps and not miss any points. · 1 year, 7 months ago

Show that $$n^{5}+n^{4}+1$$ is not prime for $$n>1$$. · 1 year, 7 months ago

$$n^{2} + n + 1$$ is factor of the given expression.Hence the result is obvious. · 1 year, 7 months ago

If $$a,b,c,d,e$$ are real numbers, prove that the roots of $$x^{5}+ax^{4}+bx^{3}+cx^{2}+dx+e=0$$ cannot all be real if $$2a^{2}<5b$$. · 1 year, 7 months ago

Let roots be $$p,q,r,s,t$$.

Sum of roots = -a & Product of roots taken two at a time = b.

Therefore Sum of squares of roots = $$a^{2} - 2b$$

If all roots are real, then by Titu's Lemma,

$$a^{2} -2b \geq \frac{a^{2}}{5}$$

$$\implies 2a^{2} \geq 5b$$

Therefore the polynomial cannot have all roots real if $$5b > 2a^{2}$$. · 1 year, 7 months ago

Let $$a,b \in R, a \neq 0$$. Show that $$a^{2}+b^{2}+\frac{1}{a^{2}}+\frac{b}{a} \ge \sqrt{3}$$. · 1 year, 7 months ago

Well the question becomes very easy if u make it a quadratic in b and then prove that the Discriminant is less than 0. I did it that way. · 1 year, 7 months ago

An easy INMO geometry problem:

Let $$ABC$$ be a triangle with circumcircle $$\Gamma$$. Let $$M$$ be a point in the interior of triangle $$ABC$$ which is also on the bisector of $$\angle A$$. Let $$AM, BM, CM$$ meet $$\Gamma$$ in $$A_{1}, B_{1}, C_{1}$$ respectively. Suppose $$P$$ is the point of intersection of $$A_{1}C_{1}$$ with $$AB$$ and $$Q$$ is the point of intersection of $$A_{1}B_{1}$$ with $$AC$$. Prove that $$PQ$$ is parallel to $$BC$$ . · 1 year, 7 months ago

Let $$ABC$$ be an acute-angled tiangle .Let O denote its circumcenter.Let $$\Pi$$ be the circle passing through the points A,O,B.Lines CA and CB meet $$\Pi$$ again at P and Q respectively.Prove that PQ is perpendicular to the line CO. · 1 year, 7 months ago

Harsh,could you please add a figure? · 1 year, 7 months ago

Hint: Let $$\overrightarrow{QP} \cap \overleftrightarrow{CO} = \{R\}$$ and $$\overleftrightarrow{CO} \cap \Pi = \{S\}$$ and then it will suffice to prove that $$\Delta CRP \sim \Delta CAS$$ · 1 year, 7 months ago

Find the least positive integer n such that 2549 divides $$n^{2545} - 2541$$ · 1 year, 7 months ago

Please post the solution , Thanks! · 1 year, 7 months ago

Let $$ABC$$ be a triangle and $$P$$ be a point inside it.Let $$x,y,z$$ denote the lengths of $$PA$$,$$PB$$,$$PC$$.Let $$a,b,c$$ be the lengths of sides $$BC$$,$$CA$$,$$AB$$.Find all points $$P$$ such that $$axy+byz+czx=abc$$. · 1 year, 7 months ago

If a circle goes through point $$A$$ of a parallelogram $$ABCD$$, cuts the two sides $$AB,AD$$ and the diagonal $$AC$$ at points $$P,R$$ and $$Q$$ respectively. Prove that $$(AP×AB)+(AR×AD)=AQ×AC$$. · 1 year, 7 months ago

I hope you all have drawn the diagram.(Consider all the segments below as vectors in the given direction.)

Let $$AS$$ be the diameter of given circle.

$$AB+AD=AC$$

Therefore taking dot product with$$AS$$

$$AB.AS+AD.AS=AC.AS$$

Using, $$a.b=|a|*|b|*cos(\theta)$$ [$$\theta$$ is angle between the vectors]

$$|AB|*|AP|+|AR|*|AD|=|AQ|*|AC|$$ · 1 year, 7 months ago

Comment deleted Dec 04, 2015

You can also use Ptolemy's theorem to the quadrilateral $$PQRA$$ and then replace the sides of $$\Delta PQR$$ by the corresponding sides of the similar triangle $$CBA$$.( $$\Delta PQR \sim \Delta CBA$$ by $$AAA$$ similarity.) · 1 year, 7 months ago

Given seven arbitrary distinct real numbers, show that there exist two numbers $$x$$ and $$y$$ such thst $$0<\dfrac{x-y}{1+xy}<\dfrac{1}{\sqrt{3}}$$. · 1 year, 7 months ago

The question is easy. Consider x and y to be some tan(A) and tan(B) respectively and consider A and B to vary from 0 to pi so that all reals are covered. now divide it into six equal portions spaced by pi/6. Now u have 6 pigeons and 7 holes and you are done. Then that expression is simply tan(A-B). · 1 year, 7 months ago

There are given $$2^{500}$$ points on a circle labeled $$1, 2, . . . , 2^{500}$$in some order. Prove that one can choose 100 pairwise disjoint chords joining some of these points so that the 100 sums of the pairs of numbers at the endpoints of the chosen chords are equal. · 1 year, 8 months ago

In the first question (sum(ω))^2>3sum(ω1*ω2) fails to hold? why? · 9 months, 2 weeks ago

@Harsh Shrivastava @Dev Sharma @Adarsh Kumar @Svatejas Shivakumar @Saarthak Marathe@Nihar Mahajan Where are u guys ?????? Please be active on the INMO practise board as well like u were on this board.... we need a lot of questions and a lot of solutions ..... · 1 year, 6 months ago

Could someone please post some nice RMO-INMO level Geometry problems ? I have nothing to work on. Please not from the previous years. · 1 year, 7 months ago

See the problems of this set. Can you suggest me some good books for Geometry? My geometry is horrible. · 1 year, 6 months ago

Thanks for that set ! For geometry I have been solving only Challenges and Thrills ... I don't have something exclusively for geometry.... · 1 year, 6 months ago

@Nihar Mahajan @Saarthak Marathe @Svatejas Shivakumar @Harsh Shrivastava @Adarsh Kumar @Dev Sharma have your respective region's RMO results been declared ? Mumbai RMO results are declared and I am selected for INMO which is on January 17. Less than a month !!!! I think we should get back to work again. Lets start an INMO practise board just like this one as it helped me a lot in my RMO preparation. Hoping for a quick response · 1 year, 7 months ago

BTW how many did you solve? · 1 year, 7 months ago

I didn't qualify :( · 1 year, 6 months ago

Hey !! That doesn't stop you from taking part in the discussions on INMO. Its okay ... Ur just in 8th standard, I came to know about RMO in 10th ...so u are at an advantage. And I am quite sure u will make it to the IMO by your 11-12th standard. I just don't know how u r able to solve olympiad questions being in 8th standard. So now bro .... u r targeting for IMO and thus u r participating in the INMO board ... · 1 year, 6 months ago

Thanks for your encouragement. I will definitely participate in the INMO board and help you reach IMO this year! · 1 year, 6 months ago

Frankly speaking even INMO is impossible for me .... but I am just enjoying math. I have got an excuse of not studying for IIT-JEE till Jan 17(INMO). · 1 year, 6 months ago

You're not an IIT-JEE aspirant? · 1 year, 6 months ago

Congratulations!! The results have not been declared from my region (I don't think that I have much chances of qualifying this time. I did many silly mistakes :( · 1 year, 7 months ago

Congo dude! I am a bit busy right now!Sorry I can't take part in the discussion now! · 1 year, 7 months ago

Congo!

Btw my region results are not out yet. · 1 year, 7 months ago

Thanks. Which is your region ? And what is the approx result date for the past few years ? · 1 year, 7 months ago

Chhattisgrah State. · 1 year, 7 months ago

All the best to everyone writing RMO! · 1 year, 7 months ago

All the best everyone ! :) · 1 year, 7 months ago

All the best everyone.!! · 1 year, 7 months ago

All the best everyone! :) · 1 year, 7 months ago

Prove that the function $$f(n)=\lfloor n+\sqrt{n}+\frac{1}{2} \rfloor$$ where $$n$$ is a positive integer, contains all numbers except perfect squares.

$$(\lfloor \rfloor)$$ represents the greatest integer function. · 1 year, 7 months ago

I have been trying this question but I have got no hint of even how to start. Could u give some starting line ? · 1 year, 7 months ago

Is n a natural number or something like that ? Because if n is any real then put n=0.5 . You will get f(0.5)=1 which is a perfect square. · 1 year, 7 months ago

yes n is a natural number. Sorry forgot to mention it · 1 year, 7 months ago

There should be no perfect squares in the domain or the range? · 1 year, 7 months ago

Range. · 1 year, 7 months ago

For any integer $$a$$, prove that $$a^{37} \equiv a \pmod{1729}$$. · 1 year, 7 months ago

Its easy just apply keep applying FLT for various factors of 1729 · 1 year, 7 months ago

Write the sum $$S_n= \displaystyle \sum_{k=0}^n{\frac{(-1)^{k}{n \choose{k}}}{k^{3}+ 9k^{2}+26k+24}}$$ in the form $$\frac{p(n)}{q(n)}$$ where $$p(n),q(n)$$ are polynomials with integer coefficients. · 1 year, 7 months ago

We first note that $$k^{3}+9k^{2}+26k+24=(k+2)(k+3)(k+4)$$.Hence, $S_n=\displaystyle \sum_{k=0}^n{\frac{(-1)^{k}{n \choose{k}}}{(k+2)(k+3)(k+4)}}$ $=\displaystyle \sum_{k=0}^n{\frac{(-1)^{k}n!}{(n-k)k!(k+2)(k+3)(k+4)}}$ $=\displaystyle \sum_{k=0}^n{\frac{(-1)^{k}(n+4)!}{(n-k!)(k+4!)}}\frac{(k+1)}{(n+1)(n+2)(n+3)(n+4)}$

Let $$T_n=S_n(n+1)(n+2)(n+3)(n+4)=\small \displaystyle \sum_{k=0}^n{(-1)^{k}{n+4 \choose{k+4}}(k+1)}$$.

We note that $$\small \displaystyle \sum_{k=0}^n{(-1)^{k}{n \choose{k}}}$$.Now,

$$\displaystyle \sum_{k=0}^n{(-1)^{k}{n \choose{k}}k}$$$$=\displaystyle \sum_{k=1}^n{(-1)^{k}n{n-1 \choose{k-1}}+0}$$$$=-n \displaystyle \sum_{j=0}^{n-1}{(-1)^{j}{n-1 \choose{j}}=0}$$.

$T_n=\displaystyle \sum_{k=0}^n{(-1)^{k}{n+4 \choose{k+4}}(k+1)}$. $=\displaystyle \sum_{k=0}^n{(-1)^{k+4}{n+4 \choose{k+4}}(k+1)}$. $=\displaystyle \sum_{k=-4}^n{(-1)^{k+4}{n+4 \choose{k+4}}(k+1)-((-3)+2(n+4)-{n+4 \choose{2}}})$.

Substituting $$j=k+4$$;

$T_n=\displaystyle \sum_{j=0}^{n+4}{(-1)^{j}{n+4 \choose{j}}(j-3)-((-3)+2(n+4)-{n+4 \choose{2}}})$.

$$T_n=\displaystyle \sum_{j=0}^{n+4}{(-1)^{j}{n+4 \choose{j}}}-3 \displaystyle \sum_{j=0}^{n+4}{(-1)^{j}{n+4 \choose{j}}(j-3)-((-3)+2(n+4)-{n+4 \choose{2}}})$$.

The first two terms are zero, hence

$$T_n={n+4 \choose{j}}-2n-8+3=\frac{(n+4)(n+3)}{2}-2n-5$$.

Hence, $$T_n=\frac{(n+1)(n+2)}{2}$$ and $$S_n=\frac{1}{2(n+3)(n+4)}$$.

Phew!! · 1 year, 7 months ago

Nicely done. There are a few mistakes in the solution but one may understand the correct part as he reads it. · 1 year, 7 months ago

A barrel contains $$2n$$ balls, numbered $$1$$ to $$2n$$. Choose three balls at random, one after the other, and with the balls replaced after each draw.

What is the probability that the three element sequence obtained has the properties that the smallest element is odd and that only the smallest element, is repeated? · 1 year, 7 months ago

1/2 · 1 year, 7 months ago

Correct. Can you show your steps? · 1 year, 7 months ago

$$a,b,c$$ are real numbers such that $$a+b+c=0$$ and $$a^2+b^2+c^2=1$$. Prove that $$a^{2}b^{2}c^{2}≤\frac{1}{54}$$. · 1 year, 8 months ago

See my note · 1 year, 7 months ago