MathDB

1

Part of 1997 IberoAmerican

Problems(2)

12nd ibmo - mexico 1997/q1.

Source: Spanish Communities

4/22/2006
Let r≥1r\geq1 be areal number that holds with the property that for each pair of positive integer numbers mm and nn, with nn a multiple of mm, it is true that ⌊nr⌋\lfloor{nr}\rfloor is multiple of ⌊mr⌋\lfloor{mr}\rfloor. Show that rr has to be an integer number.
Note: If xx is a real number, ⌊x⌋\lfloor{x}\rfloor is the greatest integer lower than or equal to xx}.
floor functioncalculusintegrationrationumber theory unsolvednumber theory
12nd ibmo - mexico 1997/q4.

Source: Spanish Communities

4/22/2006
Let nn be a positive integer. Consider the sum x1y1+x2y2+⋯+xnynx_1y_1 + x_2y_2 +\cdots + x_ny_n, where that values of the variables x1,x2,…,xn,y1,y2,…,ynx_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n are either 0 or 1.
Let I(n)I(n) be the number of 2nn-tuples (x1,x2,…,xn,y1,y2,…,yn)(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n) such that the sum of the number is odd, and let P(n)P(n) be the number of 2nn-tuples (x1,x2,…,xn,y1,y2,…,yn)(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n) such that the sum is an even number. Show that: P(n)I(n)=2n+12n−1 \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1}
combinatorics unsolvedcombinatorics