Problems

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Found: 1176

Two circles with centres A and C are tangent to each other at the point B. Two points D and E are chosen on the circles in such a way that a segment DE passes through the point B. Prove that the tangent line to one circle at the point D is parallel to the tangent line to the other circle at the point E.

Is it true that if a is a positive number, then a2a? What about a2+1a?

Consider the following sum: 11×2+12×3+13×4+ Show that no matter how many terms it has, the sum will never be larger than 1.

Is it true that if b is a positive number, then b3+b2b? What about b3+1b?

Let k be a natural number, prove the following inequality. 1k2>1k1k+1.

Show that if a is a positive number, then a3+22aa.

The numbers a, b and c are positive. By completing the square, show that a24+b2+c2abac+2bc.

Let m and n be natural numbers such that m>n. Show that: 1n2+1(n+1)2+1(n+2)2++1m2>1n1m.