MathDB

2

Part of 2008 Romania Team Selection Test

Problems(5)

Sequence and inequality

Source: Romanian TST 1 2008, Problem 2

5/1/2008
Let ai,bi a_i, b_i be positive real numbers, i\equal{}1,2,\ldots,n, n≥2 n\geq 2, such that ai<bi a_i<b_i, for all i i, and also b_1\plus{}b_2\plus{}\cdots \plus{} b_n < 1 \plus{} a_1\plus{}\cdots \plus{} a_n. Prove that there exists a c∈R c\in\mathbb R such that for all i\equal{}1,2,\ldots,n, and k∈Z k\in\mathbb Z we have (a_i\plus{}c\plus{}k)(b_i\plus{}c\plus{}k) > 0.
inequalitiesinequalities proposed
Sequences with a special property? Or not?

Source: Romanian TST 2 2008, Problem 2

6/7/2008
Are there any sequences of positive integers 1≤a1<a2<a3<… 1 \leq a_{1} < a_{2} < a_{3} < \ldots such that for each integer n n, the set \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\} contains finitely many prime numbers?
number theoryprime numbersnumber theory proposed
Counting subsets

Source: Romanian TST 4 2008, Problem 2

6/13/2008
Let m,n≥1 m, n \geq 1 be two coprime integers and let also s s an arbitrary integer. Determine the number of subsets A A of \{1, 2, ..., m \plus{} n \minus{} 1\} such that |A| \equal{} m and ∑x∈Ax≡s(modn) \sum_{x \in A} x \equiv s \pmod{n}.
modular arithmeticnumber theory proposednumber theory
Classic Fuhrmann-type configuration

Source: Romanian TST 3 2008, Problem 2

6/7/2008
Let ABC ABC be an acute triangle with orthocenter H H and let X X be an arbitrary point in its plane. The circle with diameter HX HX intersects the lines AH AH and AX AX at A1 A_{1} and A2 A_{2}, respectively. Similarly, define B1 B_{1}, B2 B_{2}, C1 C_{1}, C2 C_{2}. Prove that the lines A1A2 A_{1}A_{2}, B1B2 B_{1}B_{2}, C1C2 C_{1}C_{2} are concurrent. Remark. The triangle obviously doesn't need to be acute.
geometrygeometry proposed
Medians as diameters and a known Feuerbach-type result

Source: Romanian TST 5 2008, Problem 2

6/13/2008
Let ABC ABC be a triangle and let Ma \mathcal{M}_{a}, Mb \mathcal{M}_{b}, Mc \mathcal{M}_{c} be the circles having as diameters the medians ma m_{a}, mb m_{b}, mc m_{c} of triangle ABC ABC, respectively. If two of these three circles are tangent to the incircle of ABC ABC, prove that the third is tangent as well.
geometrygeometry proposed