MathDB

1

Part of 2005 Germany Team Selection Test

Problems(6)

Largest term in the (a+b)^n expansion

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

12/15/2004
Given the positive numbers aa and bb and the natural number nn, find the greatest among the n+1n + 1 monomials in the binomial expansion of (a+b)n\left(a+b\right)^n.
inequalitiesalgebra proposedalgebra
a Japanese Problem

Source: Japanese math Olympiad

1/15/2005
Prove that there doesn't exist any positive integer nn such that 2n2+1,3n2+12n^2+1,3n^2+1 and 6n2+16n^2+1 are perfect squares.
number theory solvednumber theory
F(xy) * f(f(y)/x) = 1

Source: 4th German TST 2005, problem 1

6/3/2005
Find all monotonically increasing or monotonically decreasing functions f:R+→R+f: \mathbb{R}_+\to\mathbb{R}_+ which satisfy the equation f(xy)⋅f(f(y)x)=1f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1 for any two numbers xx and yy from R+\mathbb{R}_+. Hereby, R+\mathbb{R}_+ is the set of all positive real numbers. Note. A function f:R+→R+f: \mathbb{R}_+\to\mathbb{R}_+ is called monotonically increasing if for any two positive numbers xx and yy such that x≥yx\geq y, we have f(x)≥f(y)f\left(x\right)\geq f\left(y\right). A function f:R+→R+f: \mathbb{R}_+\to\mathbb{R}_+ is called monotonically decreasing if for any two positive numbers xx and yy such that x≥yx\geq y, we have f(x)≤f(y)f\left(x\right)\leq f\left(y\right).
functioninequalitiesalgebra proposedalgebra
Palindromes out of two letters and coprime integers

Source: 5th German TST 2005, problem 1, not from the shortlist

5/12/2005
In the following, a word will mean a finite sequence of letters "aa" and "bb". The length of a word will mean the number of the letters of the word. For instance, abaababaab is a word of length 55. There exists exactly one word of length 00, namely the empty word. A word ww of length ℓ\ell consisting of the letters x1x_1, x2x_2, ..., xℓx_{\ell} in this order is called a palindrome if and only if xj=xℓ+1−jx_j=x_{\ell+1-j} holds for every jj such that 1≤j≤ℓ1\leq j\leq\ell. For instance, baaabbaaab is a palindrome; so is the empty word. For two words w1w_1 and w2w_2, let w1w2w_1w_2 denote the word formed by writing the word w2w_2 directly after the word w1w_1. For instance, if w1=baaw_1=baa and w2=bbw_2=bb, then w1w2=baabbw_1w_2=baabb. Let rr, ss, tt be nonnegative integers satisfying r+s=t+2r + s = t + 2. Prove that there exist palindromes AA, BB, CC with lengths rr, ss, tt, respectively, such that AB=CabAB=Cab, if and only if the integers r+2r + 2 and s−2s - 2 are coprime.
inductionLaTeXcombinatorics proposedcombinatorics
U-g-l-y problem on decimal representation of n!

Source: 6th German TST 2005, problem 1

5/30/2005
(a) Does there exist a positive integer nn such that the decimal representation of n!n! ends with the string 20042004, followed by a number of digits from the set {0;  4}\left\{0;\;4\right\} ? (b) Does there exist a positive integer nn such that the decimal representation of n!n! starts with the string 20042004 ?
number theory proposednumber theory
M / 2004 < k / n < (m + 1) / 2005

Source: 7th German TST 2005, problem 1

5/30/2005
Find the smallest positive integer nn with the following property:
For any integer mm with 0<m<20040 < m < 2004, there exists an integer kk such that
m2004<kn<m+12005.\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.
Putnaminequalitiesnumber theory unsolvednumber theory