Recently Active Olympiad Stuff Questions

Find all primes p and q ,and even numbers n > 2 , satisfying the equation pn + pn-1 + · · · + p + 1 = q2 + q + 1. ...

Not exactly Olympiad stuff, infact I don't know if it is. I just needed a section to post it. Others may try and solve it if they please -: Find the sum of terms of the GP : a+ar+ar2.......∞ where a is the value of x for wh ...

$\textbf{Solve system of equations:}\\\\ $\mathbf{x+y-z=7}$\\\\ $\mathbf{x^2+y^2-z^2=37}$\\\\ $\mathbf{x^3+y^3-z^3=1}$ ...

\hspace{-16}$If $\mathbf{\alpha\in \mathbb{R}}\;,$ Then Prove That\\\\\\ $\mathbf{\sqrt{17}\leq\sqrt{\cos^2 \alpha+4\cos \alpha +6}+\sqrt{\cos^2 \alpha-2\cos \alpha +3}\leq \sqrt{2}+\sqrt{11}}$ ...

In a regular pentagon, if the length of each side is a and that of each diagonal is b then find the numerical value of a2/b2 + b2/a2 Note : you are not allowed to use trigonometry in this..!! maybe complex can help ...

The no. of ways of selecting n things out of 3n things of which n are of one kind and alike and n are of 2nd kind and rest are unlike is: ...

Prove that if n>2,then there exists a prime p satisfying n>p>n! ...

*Image* just focus... its simple.. !! ...

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*Image* sine rule can help too... ...

Given four points A_{1}, A_{2}, A_{3}, A_{4} in the plane, no three collinear, such that \ A_{1}A_{2}\cdot A_{3}A_{4}= A_{1}A_{3}\cdot A_{2}A_{4}= A_{1}A_{4}\cdot A_{2}A_{3} denote by O_i the circumcenter of \triangle A_{j}A_ ...

$\textbf{If $\mathbf{a,b,c>0}$ and $\mathbf{a^{2009}+b^{2009}+c^{2009}=3}.$ Then Find Max. and\\\\ Min. value of $\mathbf{a^4+b^4+c^4}$} ...

http://students.iitk.ac.in/takneek/media/nuq2m.pdf ...

Prove that for all real numbers x,y , |cos(x)|+ |cos(y)| + |cos(x+y)|\geq 1 ...

$\textbf{In a $\mathbf{\triangle ABC},$ Prove that }$\\\\ \mathbf{\frac{1+Cos\;B\cdot Cos\;C}{1+cos \;A}+\frac{1+Cos\;C\cdot Cos\;A}{1+cos \;B}+\frac{1+Cos\;A\cdot Cos\;B}{1+cos \;C}\geq \frac{5}{2}}$ ...

1)Find all function from +ve reals to the same satisfying f(f(x)-x) = 2x for all x>0 2) f is a function from the set of natural numbers to the same. satisfying f(n+1)>f(f(n)) for all natural number n. Prove that f(n) = ...

A triangle has all its sides as integers marked l,m,n with l>m>n. Also given that \left\{\frac{3^l}{10^4} \right\}=\left\{\frac{3^m}{10^4} \right\}=\left\{\frac{3^n}{10^4} \right\} Find the minimum perimeter of the tria ...

find all integers a such that (x+a)(x+1991) + 1 can be written as (x+b)(x+c) where b and c are integers themselves. it implies that we have to find all integers a , b and c such that 1991 + a = b + c and 1 + 1991a = bc for so ...

Two of the squares of a 7X7 checkboard are painted yellow. and the rest are painted green..two color schemes are said to be equivalent if one can be obtained from the other by a rotation of the board...how many inequivalent s ...

In a mathematical competition 6 problems were posed to the contestants. Each pair of problems was solved by more than 2 5 of the contestants. Nobody solved all 6 problems. Show that there were at least 2 contestants who each ...

If difference of cubes of two consecutive natural numbers is equal to n2 (n is a positive integer). Prove that 2n-1 is a perfect square. this is not very difficult ...

\hspace{-15}$\textbf{Find Integer solution for}\\\\ $\mathbf{x(x+1)(x+2)(x+3)(x+4)(x+5)=y^2-1}$ ...

Dear friends The contest session of IMO 2011 starts tomorrow Lets root for our team. You can check them out http://official.imo2011.nl/year_reg_team.aspx?year=2011&code=IND. Its great to see a girl member, Mrudul Thatte on th ...

prove that for all integers, n > 7 C (n,7) - [n/7] is divisible by 7. here [.] denotes the greatest integer less then or equal to the number in the bracket. i tried it but all i could get was that for values of n in [7,13] ...

solve for x . (x2 + x -3)3 + (2x2 - x - 1)3 = 27(x2 - 1)3 ...

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Let " x " and " y " be positive integers such that " xy " divides " x 2 + y 2 + 1 " . Prove that - x 2 + y 2 + 1/xy = 3 . I will add some more if I see that people are taking interest in this . ...

Let N0 denote the set of nonnegative integers. Find all functions from N0 to itself such that. \ f(m+f(n)) = f(f(m))+f(n)\qquad\text{for all}\; m, n\in\mathbb{N}_{0}. \ ...

1. In a race course there are 5 tracks .In how many minimum number of chances can you find out the fastest three horses, if there were 25 horses initially. 2. Can you use all 9 numerals - 1, 2, 3, 4, 5, 6, 7, 8 and 9 - above ...

problem:Find all function G from the set of natural numbers to natural numbers such that (g(m) +n)(g(n)+m) is a perfect square for all m,n (natural numbers) . solution: Lets take g(m) = m+ f(m) The given expression equals (m+ ...