MathDB

2

Part of 2004 Bulgaria Team Selection Test

Problems(4)

Primes and square-free integers

Source: Bulgarian IMO TST 2004, Day 1, Problem 2

7/8/2013
Find all primes p≥3p \ge 3 such that p−⌊p/q⌋qp- \lfloor p/q \rfloor q is a square-free integer for any prime q<pq<p.
floor functionnumber theory proposednumber theory
Coinciding orthocenters

Source: Bulgarian IMO TST 2004, Day 2, Problem 2

7/8/2013
Let HH be the orthocenter of △ABC\triangle ABC. The points A1≠AA_{1} \not= A, B1≠BB_{1} \not= B and C1≠CC_{1} \not= C lie, respectively, on the circumcircles of △BCH\triangle BCH, △CAH\triangle CAH and △ABH\triangle ABH and satisfy A1H=B1H=C1HA_{1}H=B_{1}H=C_{1}H. Denote by H1H_{1}, H2H_{2} and H3H_{3} the orthocenters of △A1BC\triangle A_{1}BC, △B1CA\triangle B_{1}CA and △C1AB\triangle C_{1}AB, respectively. Prove that △A1B1C1\triangle A_{1}B_{1}C_{1} and △H1H2H3\triangle H_{1}H_{2}H_{3} have the same orthocenter.
geometrycircumcirclegeometric transformationreflectionrotationtrapezoidpower of a point
Graph theory

Source: Bulgarian IMO TST 2004, Day 3, Problem 2

7/8/2013
The edges of a graph with 2n2n vertices (n≥4n \ge 4) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with nn vertices. Find the least possible number of blue edges.
combinatorics proposedcombinatoricsRamsey Theorygraph theory
Inequality involving square roots

Source: Bulgarian IMO TST 2004, Day 4, Problem 2

7/8/2013
Prove that if a,b,c≥1a,b,c \ge 1 and a+b+c=9a+b+c=9, then ab+bc+ca≤a+b+c\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}
inequalitiesalgebraInequality