MathDB

10

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #10

Source:

12/26/2011
There exists a polynomial PP of degree 55 with the following property: if zz is a complex number such that z5+2004z=1z^5+2004z=1, then P(z2)=0P(z^2)=0. Calculate the quotient P(1)P(−1)\tfrac{P(1)}{P(-1)}.
algebrapolynomialdifference of squaresspecial factorizations
2004 Calculus #10

Source:

11/29/2011
Let P(x)=x3−32x2+x+14P(x)=x^3-\tfrac{3}{2}x^2+x+\tfrac{1}{4}. Let P[1](x)=P(x)P^{[1]}(x)=P(x), and for n≥1n\ge1, let Pn+1(x)=P[n](P(x))P^{n+1}(x)=P^{[n]}(P(x)). Evaluate: ∫01P[2004](x) dx. \displaystyle\int_{0}^{1} P^{[2004]} (x) \ \mathrm{d}x.
calculusintegrationtrigonometryinduction
2004 Combinatorics #10

Source:

12/31/2011
In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the two edge cards (so the configuration after the first move is either RRQ or QRR). What is the probability that, after 2004 moves, the center card is the queen?
probability
2004 HMMT Geometry # 10

Source:

3/3/2024
Right triangle XYZXY Z has right angle at YY and XY=228XY = 228, YZ=2004Y Z = 2004. Angle YY is trisected, and the angle trisectors intersect XZXZ at PP and QQ so that XX, PP, QQ,ZZ lie on XZXZ in that order. Find the value of (PY+YZ)(QY+XY)(PY + Y Z)(QY + XY ).
geometry
2004 General, part 1 #10

Source:

3/8/2024
A floor is tiled with equilateral triangles of side length 11, as shown. If you drop a needle of length 22 somewhere on the floor , what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle — touching along edges or at corners doesn't qualify.) https://cdn.artofproblemsolving.com/attachments/5/6/e7555c22ffe890b46a3ebdbda2169d23e43700.png
combinatorics
2004 General, part 2 #10

Source:

3/8/2024
A lattice point is a point whose coordinates are both integers. Suppose Johann walks in a line from the point (0,2004)(0, 2004) to a random lattice point in the interior (not on the boundary) of the square with vertices (0,0)(0, 0), (0,99)(0, 99), (99,99)(99,99), (99,0)(99, 0). What is the probability that his path, including the endpoints, contains an even number of lattice points?
combinatorics