Recently Active Olympiad Stuff Questions

Q Prove that the blue part is equal to the yellow part It is a square and the triangles have the intersection at the mid point of the square *Image* ...

Find all positive integral solutions (a,b,c) to the equation (a+1)(b+1)(c+1)=2(abc + 1) ...

Find all positive integral solutions (a,b,c) to the equation (a+1)(b+1)(c+1)=2(abc + 1) ...

Let a and b be two positive integers such that for any positive integer n, an + n divides bn + n. Prove that a = b. ...

n is a positive integer. Prove that we cannot find an integer m and an integer r > 1 such that n(n + 1)(n + 2) = mr. ...

determine all pairs (h,s) of positive integers with the following property if one draws s horizontal lines and another s lines which satisfy 1)they r not horizontal 2)no two of them are parallel 3)no three of h+s are concurre ...

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LET \left(a_{in} \right)_{n\geq 1} i\epsilon \left\{1,2,3,.,p \right\} sequences of strictly real pozitive numbers. Prove that: ( \sum_{k=1}^{n} \frac{1}{ \prod_{i=1}^{p} a_{ik} } ) ( \sum_{k=1}^{n}( \sum_{i=1}^{p} a_{ik} )^{ ...

Book: Inegalitati; Authors:L.Panaitopol,V. Bandila,M.Lascu i already gave one more from this book Let x,yε R. Prove that if m and n ε N and of the same parity we have: a] \frac{x^m+y^m}{2} \frac{x^n+y^n}{2} \leq \frac{x^{m+ ...

In triangle ABC, D is a point on AB and E is a point on AC such that BE and CD are bisectors of ∠B and ∠C respectively. Let Q,M and N be the feet of perpendiculars from the midpoint P of DE onto BC, AB and AC, respectivel ...

A town has several clubs. Given any two residents there is exactly one club that both belong to. Given any two clubs, there is exactly one resident who belongs to both. Each club has at least 3 members. At least one club has ...

There are 2 players in this game. The first player starts, and only positive integers can be written down on a piece of paper. If the numbers a,b have been written on the piece of paper any number written after those two cann ...

there are 1000 men standing in a circle numbere 1 to 1000 on their back. the person with number 1 is given a sword. he kills 2 and gives sword to 3. 3 kills 4 and passes the sword to 5. the process continues so on............ ...

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suppose there are 997 points given in a plane. if every two points are joined by a line segment with its mid point coloured in red, show that there are atleast 1991 red points in the plane. can u find a special case with exac ...

Check this question i get from harward website. I like the solution. Please try. Becasue I have casue this mathematician to leave, i will give some good question now :) :D *Image* ...

Prove that for any integer n n7+7 is not a perfect square i have proved it for all cases except for n of the form - 7k+1 plz help . ...

which is greater? 99^n + 100^n or 101^n??? If you get ambiguous answers then specify the intervals for which 99^n+100^n is greater than 101^n or less than it. ...

Let p,q,r,be three prime numbers such that 5<=p<q<r and 2p2-r2>=49 2q2-r2<=193 find p, q ,r ...

Find the equation of the parabola whose axis is parallel to y-axis and which passes through the points (0,4) (1,9) (-2,6) and determine its latus rectum.??? ...

Prove that there exist two infinite sequences (an)n≥1 and (bn)n≥1 of positive integers such that the following conditions hold simultaneously: (i) 1 < a1 < a2 < a3 .... (ii) an<bn<a2n, for all n ≥1 ( ...

find all functions f:R->R such that for all x,y belonging to - R f(f(x)+y)=f(x2-y) + 4yf(x) this has a two line answer!! ...

Lakshya: 7:17pm 14-11-08 The question is that there is a right angled triangle ABC right angled at B. The angle bisectors of angle A and C meet at a point P. From P a perpendicular PM is drawn to AC. If the length of PM is 4r ...

Suppose a and b are real numbers such that the roots of the cubic equation ax3-x2+bx-1 = 0 are all positive real numbers. Prove that : (i) 0 < 3ab ≤1 (ii) b≥ 3 P.S. I can't believe I missed this ... this one i ...

Three non-zero real numbers a,b,c are said to be in H.P. if 1/a + 1/c = 2/b. Find all three-term H.P. a,b,c of strictly increasing positive integers in which a = 20 and b divides c. ...

Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter 2008. ...

Find the number of all 6-digit natural numbers such that the sum of their digits is 10 and each of the digits 0,1,2,3 occurs at least once in them. ...

P.T . n4 + 4n is not a prime , for all integral values of n. ...

1/2+1/3+1/4.....+1/n is never an integer.. prove Hint: think of the largest prime! ...

Show dat there is no real constant c>0 such dat cos√(x+c)=cos√x for real no.s x ≥ 0. ...