MathDB

5

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 HMMT Algebra #5: Maximum Possible Coefficient

Source:

2/16/2013
Let aa and bb be real numbers, and let rr, ss, and tt be the roots of f(x)=x3+ax2+bx−1f(x)=x^3+ax^2+bx-1. Also, g(x)=x3+mx2+nx+pg(x)=x^3+mx^2+nx+p has roots r2r^2, s2s^2, and t2t^2. If g(−1)=−5g(-1)=-5, find the maximum possible value of bb.
HMMTquadratics
2013 HMMT Team #5: Homogeneously Divisible by 2013

Source:

3/27/2013
Thaddeus is given a 2013×20132013 \times 2013 array of integers each between 11 and 20132013, inclusive. He is allowed two operations:
1. Choose a row, and subtract 11 from each entry.
2. Chooses a column, and add 11 to each entry.
He would like to get an array where all integers are divisible by 20132013. On how many arrays is this possible?
HMMT
2013 HMMT Guts #5: Rahul and his Cards

Source:

3/26/2013
Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.
HMMT
2013 HMMT Geometry # 5

Source:

3/3/2024
In triangle ABCABC , ∠A=45o\angle A = 45^o and MM is the midpoint of BC‾\overline{BC}. AM‾\overline{AM} intersects the circumcircle of ABCABC for the second time at DD, and AM=2MDAM = 2MD. Find cos∠AODcos\angle AOD, where OO is the circumcenter of ABCABC.
geometry