MathDB

1:

Part of 1994 Romania TST for IMO

Problems(3)

"odd" sets and "even" sets

Source: Romania TST 1994

8/28/2009
Let X_n\equal{}\{1,2,...,n\},where n≥3 n \geq 3. We define the measure m(X) m(X) of X⊂Xn X\subset X_n as the sum of its elements.(If |X|\equal{}0,then m(X)\equal{}0). A set X⊂Xn X \subset X_n is said to be even(resp. odd) if m(X) m(X) is even(resp. odd). (a)Show that the number of even sets equals the number of odd sets. (b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets. (c)Compute the sum of the measures of the odd sets.
combinatorics proposedcombinatorics
Find the smallest nomial

Source: Romanian team selection test 1994, 3rd exam, problem 1

10/19/2005
Find the smallest nomial of this sequence that a1=199319941995a_1=1993^{1994^{1995}} and an+1={an2if n is evenan+7if n is odd. a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases}
quadraticsmodular arithmeticnumber theory proposednumber theory
Romania tst

Source: Romania TST for IMO 1994,third test

9/6/2017
Let pp be a (positive) prime number. Suppose that real numbers a1,a2,...,ap+1a_1, a_2, . . ., a_{p+1} have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.
algebranumber theory