MathDB

2

Part of 2015 Romania Team Selection Tests

Problems(5)

Triangle inequality

Source: Romania TST 2015 Day 1 Problem 2

4/9/2015
Let ABCABC be a triangle, and let rr denote its inradius. Let RAR_A denote the radius of the circle internally tangent at AA to the circle ABCABC and tangent to the line BCBC; the radii RBR_B and RCR_C are defined similarly. Show that 1RA+1RB+1RC≤2r\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}.
circlestriangle inequalityinradiusinequalitiesgeometry
Concentric circles !

Source: Romania TST 2015 Day 2 Problem 2

6/4/2015
Let ABCABC be a triangle . Let A′A' be the center of the circle through the midpoint of the side BCBC and the orthogonal projections of BB and CC on the lines of support of the internal bisectrices of the angles ACBACB and ABCABC , respectively ; the points B′B' and C′C' are defined similarly . Prove that the nine-point circle of the triangle ABCABC and the circumcircle of A′B′C′A'B'C' are concentric.
Nine-point circlecirclesSupportgeometrycircumcircle
Bounded sequence !!

Source: Romania TST 2015 Day 3 Problem 2

6/4/2015
Let (an)n≥0(a_n)_{n \geq 0} and (bn)n≥0(b_n)_{n \geq 0} be sequences of real numbers such that a0>12 a_0>\frac{1}{2} , an+1≥ana_{n+1} \geq a_n and bn+1=an(bn+bn+2)b_{n+1}=a_n(b_n+b_{n+2}) for all non-negative integers nn . Show that the sequence (bn)n≥0(b_n)_{n \geq 0} is bounded .
boundedSequencesRecurrencealgebraRomanian TST
Maximal number of divisors a binomial coefficient has !!!

Source: Romania TST 2015 Day 4 Problem 2

6/4/2015
Given an integer k≥2k \geq 2, determine the largest number of divisors the binomial coefficient (nk)\binom{n}{k} may have in the range n−k+1,…,nn-k+1, \ldots, n , as nn runs through the integers greater than or equal to kk.
binomial coefficientsRomanian TSTnumber theory
Air companies !!

Source: Romania TST 2015 Day 5 Problem 2

6/4/2015
Let nn be an integer greater than 11, and let pp be a prime divisor of nn. A confederation consists of pp states, each of which has exactly nn airports. There are pp air companies operating interstate flights only such that every two airports in different states are joined by a direct (two-way) flight operated by one of these companies. Determine the maximal integer NN satisfying the following condition: In every such confederation it is possible to choose one of the pp air companies and NN of the npnp airports such that one may travel (not necessarily directly) from any one of the NN chosen airports to any other such only by flights operated by the chosen air company.
graph theorycombinatoricsRomanian TST