Recently Active Olympiad Stuff Questions

Prove that among the +ve numbers a,2a,3a,...,(n-1)a, there is one that differs by an integer by at most 1/n ...

Let p(x) = x2+ax+b be a quadratic polynomial. 'a' and 'b' belongs to integers. Prove that for any integer 'n' ,there is some 'm' such that P(n)P(n+1) = P(m) Does this question have any good solution? one can show that m = an ...

prove that every integer can be expressed in this form and also for a partical integer there exists unique values of a and b n=1/2(a+b-1)(a+b-2) +a my sol i solved it involes something related to consecutive numbers and remai ...

IN A TRIANGLE ABC, ALTITUDE FROM C ON AB AT F MEASURES 8 UNITS AND AB EQUALS 6 UNITS. IF M AND P ARE THE MID POINTS OF AF AND BC. FIND THE LENGTH OF PM. ...

IN A RIGHT TRIANGLE ACB, RIGHT ANGLED AT C ANGLE BISECTORS OF A AND B MEET BC AND AC AT P AND Q RESPECTIIVELY. M AND N ARE FEET OF PERPENDICULARS FROM P AND Q ON AB. FIND ANGLE MCN. ...

1.Consider a circle with centre O.2 chords AB and CD extended intersect at a point P outside the circle.If <AOC=43 and <BPD=18,then the value of <BOD is: a.36 b.29 c.7 d.25 ...

Explain Lagranges Multipliers with some examples (please ;) atleast giv some q on lagranges multipliers (related to inequality of course) ...

3 . *Image* " a r , b r and c r" are three sequences of real numbers . ...

(From Brahmagupta, 7th century A.D.) A girl was carrying a basket of eggs, and a man driving a horse hit the basket and broke all the eggs. Wishing to pay for the damage, he asked the girl how many eggs there were. The girl s ...

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Prove the following two inequalities : - 1 . *Image* Please note that " a r and b r " are two sequences of real numbers . 2 . *Image* Also infer when the two equalities hold . Note : - The first problem has got many solutions ...

Find all continuously differentiable function f ,with the property that f(x) > 0 and f(f(x)) = f'(x) for all x ε R Edits: got this one. couldn't do it yesterday,believe me;) it should be moved to the calculus section. ...

Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O. Suppose the area of traingle OAF is the geometric mean of those of OAB and OEF; and the area of triangle OBC is the geometric mean of those ...

Solve the equation y3= x3 + 8x2 - 6x + 8 for positive integers. I have a method which is not very good. ...

1.Let a1=b1=1, a2=2, b2=3, an+1=an+bn, bn+1=2an+bn, then: A> bn2=2an2+1, if nis odd. B> bn2=2an2-1, if n is odd. C> limn->∞ bn/an= 2 -1 D> limn->∞ bn/an= 2 (Multiple correct) ...

Find all primes p and q such that p2+ 7pq + q2 is square of an integer. (dbt) ...

Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer. ...

Find the number of 4-digit numbers(in base 10) having non-zero digits and which are divisible by 4 but not by 8. ...

1. if a1 , a2 , a3 ... a7 are seven not necessarily distinct real numbers such that 1<ai<13 , prove that we can construct a triangle with its sides with length ai 2 If Nn denote the nth son square integer so that N1=2 , ...

please tell me about any olympiads which undergraduates may appear for. ...

1.N is a 50 digit no.All the digits except the 26th from the right are 1.If N is divisible by 13 then the unknown digit is? ...

In an isocelous right-angled triangle ABC, right angled at A, M and N are two points on BC such that BM2 + CN2 = MN2 then prove that, LMAN = 45° ...

There are 5 pairs of shoes in a cupboard from which 4 shoes are picked at random. The probability that there is atleast one pair is: (a) 8/21 (b) 11/21 (c) 12/21 (d) 13/21 Please explain if you know the solution... Thank you! ...

*Image* ABC is a an acute-angled trianlge in which LB = 2LC and AD is the internal bisector of A which meets BC at D such that, CD = AB prove that, LA = 72° An interesting problem no doubt but an easy one though..!! Its from ...

Let Fm be the m-th Fibonacci number given by F1 = F2=1 and Fm+2 = Fm+1 + Fm for all m≥1 Show that summation of C(n,k) = Fm+1 Here the above sum is over all pairs of integers n≥k≥0 with n+ k = m ...

*Image* Please discuss the questions Got the answer of Q3. as 837 Q4. {1,2,4} and Q6. 43 ...

Please explain the stewart's theorem. ...

My Friend "Mr G is" stuck at the upper-left corner of a 4x4 grid. Every second, he takes a step to an adjacent square (no diagonal movements allowed) with equal probability. Immediately after he changes squares, there is a 1/ ...

How many questions did you guys crack? Post answers of relevant questions. Expected cutoff for 12th? ...

Let Xn be the n-th non square positive integer. Thus X1=2 , X2=3 , X3=5 , X4=6 , etc. For a positive number x, denote integer closest to it by <x> , if x = m + 0.5 where m is an integer then define <x> by m , for ...