MathDB

3

Part of 2005 Germany Team Selection Test

Problems(7)

Much simpler than it looks

Source: 1st German pre-TST 2005, 6 Dec 2004, Problem 3

12/15/2004
Let aa, bb, cc, dd and nn be positive integers such that 7⋅4n=a2+b2+c2+d27\cdot 4^n = a^2+b^2+c^2+d^2. Prove that the numbers aa, bb, cc, dd are all ≥2n−1\geq 2^{n-1}.
number theory proposednumber theory
in-radius, ex-radii

Source: 3rd German pre-TST 2005, 12 Feb 2005, Problem 3

2/24/2005
Let ABC be a triangle and let r,ra,rb,rcr, r_a, r_b, r_c denote the inradius and ex-radii opposite to the vertices A,B,CA, B, C, respectively. Suppose that a>ra,b>rb,c>rca>r_a, b>r_b, c>r_c. Prove that (a) △ABC\triangle ABC is acute. (b) a+b+c>r+ra+rb+rca+b+c > r+r_a+r_b+r_c.
geometryinradiustrigonometryinequalitiesfunctiontriangle inequalitygeometry solved
Exercise your mental arithmetic

Source: 3rd German IMO TST 2005, problem 3

5/17/2005
A positive integer is called nice if the sum of its digits in the number system with base 3 3 is divisible by 3 3. Calculate the sum of the first 2005 2005 nice positive integers.
number theory proposednumber theory
Set of divisors of a recurrent sequence

Source: 4th German TST 2005, problem 3

6/3/2005
Let bb and cc be any two positive integers. Define an integer sequence ana_n, for n≥1n\geq 1, by a1=1a_1=1, a2=1a_2=1, a3=ba_3=b and an+3=ban+2an+1+cana_{n+3}=ba_{n+2}a_{n+1}+ca_n. Find all positive integers rr for which there exists a positive integer nn such that the number ana_n is divisible by rr.
inductionnumber theory proposednumber theory
Classical and weak lower bound for AP + BP + CP

Source: 5th German TST 2005, problem 3, not from the shortlist

5/12/2005
Let ABCABC be a triangle with area SS, and let PP be a point in the plane. Prove that AP+BP+CP≥234SAP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}.
geometryinequalitiesgeometry proposed
9 HS^2 + 4 (AH * AI + BH * BI + CH * CI) >= 3d^2

Source: 6th German TST 2005, problem 3

5/30/2005
Let ABCABC be a triangle with orthocenter HH, incenter II and centroid SS, and let dd be the diameter of the circumcircle of triangle ABCABC. Prove the inequality 9⋅HS2+4(AH⋅AI+BH⋅BI+CH⋅CI)≥3d2,9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2, and determine when equality holds.
geometryincentercircumcircleinequalitiesinradiusEulervector
the sum of p numbers is divisible by p

Source: Erdos-Ginzburg-Zif theorem

3/5/2005
We have 2p−12p-1 integer numbers, where pp is a prime number. Prove that we can choose exactly pp numbers (from these 2p−12p-1 numbers) so that their sum is divisible by pp.
inequalitiesmodular arithmeticalgorithminductionnumber theorynumber theory unsolved