MathDB
1986 MMPC , Part 2 = Michigan Mathematics Prize Competition

Source:

October 18, 2022
MMPCalgebrageometrycombinatoricsnumber theory

Problem Statement

p1. △DEF\vartriangle DEF is constructed from equilateral △ABC\vartriangle ABC by choosing DD on ABAB, EE on BCBC and FF on CACA so that DBAB=ECBC=FACA=a\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a, where aa is a number between 00 and 1/21/2. (a) Show that △DEF\vartriangle DEF is also equilateral. (b) Determine the value of aa that makes the area of △DEF\vartriangle DEF equal to one half the area of △ABC\vartriangle ABC.
p2. A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly 11 red ball and 2 2 white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly 19861986 red balls and 19861986 white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color.
p3. Let a,ba, b, and cc be three consecutive positive integers, with a<b<c.a < b < c. (a) Show that abab cannot be the square of an integer. (b) Show that acac cannot be the square of an integer. (c) Show that abcabc cannot be the square of an integer.
p4. Consider the system of equations x+y=2\sqrt{x}+\sqrt{y}=2 x2+y2=5 x^2+y^2=5 (a) Show (algebraically or graphically) that there are two or more solutions in real numbers xx and yy. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection.
p5. Let nn and mm be positive integers. An n×mn \times m rectangle is tiled with unit squares. Let r(n,m)r(n, m) denote the number of rectangles formed by the edges of these unit squares. Thus, for example, r(2,1)=3r(2, 1) = 3. (a) Find r(2,3)r(2, 3). (b) Find r(n,1)r(n, 1). (c) Find, with justification, a formula for r(n,m)r(n, m).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Back to ProblemsView on AoPS