MathDB

10

Part of 1999 Harvard-MIT Mathematics Tournament

Problems(6)

1999 Advanced Topics #10: Minimum Possible Value

Source:

6/21/2012
Find the minimum possible value of the largest of xyxy, 1−x−y+xy1-x-y+xy, and x+y−2xyx+y-2xy if 0≤x≤y≤10\leq x \leq y \leq 1.
1999 Algebra #10: Tunnels Through a Pyramid

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6/21/2012
Pyramid EARLYEARLY is placed in (x,y,z)(x,y,z) coordinates so that E=(10,10,0),A=(10,−10,0)E=(10,10,0),A=(10,-10,0), R=(−10,−10,0)R=(-10,-10,0), L=(−10,10,0)L=(-10,10,0), and Y=(0,0,10)Y=(0,0,10). Tunnels are drilled through the pyramid in such a way that one can move from (x,y,z)(x,y,z) to any of the 99 points (x,y,z−1)(x,y,z-1), (x±1,y,z−1)(x\pm 1,y,z-1), (x,y±1,z−1)(x,y\pm 1, z-1), (x±1,y±1,z−1)(x\pm 1, y\pm 1, z-1). Sean starts at YY and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability 19\dfrac{1}{9}. What is the probability that he ends up at the point (8,9,0)(8,9,0)?
geometry3D geometrypyramidanalytic geometryprobability
1999 Calculus #10: Inscribing and Circumscribing to Infinity

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6/21/2012
Let AnA_n be the area outside a regular nn-gon of side length 11 but inside its circumscribed circle, let BnB_n be the area inside the nn-gon but outside its inscribed circle. Find the limit as nn tends to infinity of AnBn\dfrac{A_n}{B_n}.
calculusgeometrycircumcircle
1999 HMMT Geometry # 10

Source:

3/3/2024
In the figure below, AB=15AB = 15, BD=18BD = 18, AF=15AF = 15, DF=12DF = 12, BE=24BE = 24, and CF=17CF = 17. Find BG:FGBG : FG. https://cdn.artofproblemsolving.com/attachments/9/e/dc171c52961442f9846d2fce858937ff9fb7e8.png
geometry
1999 HMMT Team #10 combo geo with 5 points at lattice points

Source:

3/8/2024
If 55 points are placed in the plane at lattice points (i.e. points (x,y)(x,y) where xx and yy are both integers) such that no three are collinear, then there are 1010 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than 1/21/2?
geometrycombinatoricscombinatorial geometry
1999 HMMT Oral #10

Source:

3/8/2024
A,B,C,D,A, B, C, D, and EE are relatively prime integers (i.e., have no single common factor) such that the polynomials 5Ax4+4Bx3+3Cx2+2Dx+E5Ax^4 +4Bx^3 +3Cx^2 +2Dx+E and 10Ax3+6Bx2+3Cx+D10Ax^3 +6Bx^2 +3Cx+D together have 77 distinct integer roots. What are all possible values of AA? Your team has been given a sealed envelope that contains a hint for this problem. If you open the envelope, the value of this problem decreases by 20 points. To get full credit, give the sealed envelope to the judge before presenting your solution.
algebra