MathDB

4

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 HMMT Algebra #4: All Possible values of A

Source:

2/16/2013
Determine all real values of AA for which there exist distinct complex numbers x1x_1, x2x_2 such that the following three equations hold: \begin{align*}x_1(x_1+1)&=A\\x_2(x_2+1)&=A\\x_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*}
HMMTcomplex numbers
2013 HMMT Guts #4: Spencer and his Burritos

Source:

3/26/2013
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?
HMMTcomplementary counting
2013 Team #4: Sum of real numbers

Source:

2/18/2013
Let a1a_1, a2a_2, a3a_3, a4a_4, a5a_5 be real numbers whose sum is 2020. Determine with proof the smallest possible value of ∑1≤i≤j≤5⌊ai+aj⌋. \displaystyle\sum_{1\le i \le j \le 5} \lfloor a_i + a_j \rfloor.
floor function
2013 HMMT Geometry # 4

Source:

3/3/2024
Let ω1\omega_1 and ω2\omega_2 be circles with centers O1O_1 and O2O_2, respectively, and radii r1r_1 and r2r_2, respectively. Suppose that O2O_2 is on ω1\omega_1. Let AA be one of the intersections of ω1\omega_1 and ω2\omega_2, and BB be one of the two intersections of line O1O2O_1O_2 with ω2\omega_2. If AB=O1AAB = O_1A, find all possible values of r1r2\frac{r_1}{r_2} .
geometry