MathDB

1

Part of 2010 Slovenia National Olympiad

Problems(4)

Find number t for which 2a^2-3abt+b^2=2a^2+abt-b^2=0

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11/14/2010
For a real number tt and positive real numbers a,ba,b we have 2a2−3abt+b2=2a2+abt−b2=02a^2-3abt+b^2=2a^2+abt-b^2=0 Find t.t.
algebra proposedalgebra
Find the value of the expression in fractions[Slovenia 2010]

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11/14/2010
Let a,ba,b be real numbers such that ∣a∣≠∣b∣|a| \neq |b| and a+ba−b+a−ba+b=6.\frac{a+b}{a-b}+\frac{a-b}{a+b}=6. Find the value of the expression a3+b3a3−b3+a3−b3a3+b3.\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.
algebra proposedalgebra
a^2+b^2+c^2 is divisible by 4 [Slovenia 2010]

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11/14/2010
Let a,b,ca,b,c be positive integers. Prove that a2+b2+c2a^2+b^2+c^2 is divisible by 44, if and only if a,b,ca,b,c are even.
modular arithmeticnumber theorynumber theory proposed
Primes p,q,r for which p-q,p-r,q-r are prime [Slovenia 2010]

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11/16/2010
Find all prime numbers p,qp, q and rr such that p>q>rp>q>r and the numbers p−q,p−rp-q, p-r and q−rq-r are also prime.
number theoryprime numbersnumber theory proposed