9.3

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

min (a+b)(b+c)/(a+2b+c), 1<=a,b,c<=2 (I Soros Olympiad 1994-95 R1 9.3 10.2)

Source:

Find the smallest possible value of the expression

\frac{(a+b) (b + c)}{a + 2b+c}

a, b, c

are arbitrary numbers from the interval

[1,2]

algebrainequalitiesmin

equal quads (I Soros Olympiad 1994-95 R2 9.3)

Source:

Is there a quadrilateral in which the position of any vertex can be changed, leaving the other three in place, so that the resulting four points serve as the vertices of a quadrilateral equal to the original one?

a fly in a 1995 x 1995 chessboard

Source: : I Soros Olympiad 1994-95 Ukraine R2 9.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

Given is a square board measuring

1 995 \times 1 995

. These cells are painted with black and white paints in a checkerboard pattern, so that the corner cells are black. A spider sitting on one of the black cells can crawl to the cell on the same side as the one it occupies in one step. Prove that a spider can always reach a fly sitting motionless in another black cell by visiting all the cells of the board once.