MathDB

3

Part of 2005 All-Russian Olympiad

Problems(3)

ARO 2005 inequality

Source: ARO 2005 - problem 9.3

4/30/2005
Given three reals a1, a2, a3>1, S=a1+a2+a3a_1,\,a_2,\,a_3>1,\,S=a_1+a_2+a_3. Provided ai2ai−1>S{a_i^2\over a_i-1}>S for every i=1, 2, 3i=1,\,2,\,3 prove that 1a1+a2+1a2+a3+1a3+a1>1.\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\frac{1}{a_3+a_1}>1.
inequalitiesalgebra unsolvedalgebra
the midpoint of PQ lies on A'B'

Source: ARO 2005 - 10.6 / 9.7

4/30/2005
We have an acute-angled triangle ABCABC, and AA′,BB′AA',BB' are its altitudes. A point DD is chosen on the arc ACBACB of the circumcircle of ABCABC. If P=AA′∩BD,Q=BB′∩ADP=AA'\cap BD,Q=BB'\cap AD, show that the midpoint of PQPQ lies on A′B′A'B'.
geometrycircumcircletrigonometryratioparallelogramgeometric transformationhomothety
minimal number of questions necessary to find all numbers

Source: ARO 2005 - problem 10.3 / 11.2

4/30/2005
Given 2005 distinct numbers a1, a2,…,a2005a_1,\,a_2,\dots,a_{2005}. By one question, we may take three different indices 1≤i<j<k≤20051\le i<j<k\le 2005 and find out the set of numbers {ai, aj, ak}\{a_i,\,a_j,\,a_k\} (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers aia_i.
ceiling functioninductioncombinatorics unsolvedcombinatorics