MathDB

5

Part of 2023 Stanford Mathematics Tournament

Problems(3)

SMT 2023 Algebra #5

Source:

5/3/2023
Suppose α,β,γ∈{−2,3}\alpha,\beta,\gamma\in\{-2,3\} are chosen such that M=max⁡x∈Rmin⁡y∈R≥0αx+βy+γxyM=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy is finite and positive (note: R≥0\mathbb{R}_{\ge0} is the set of nonnegative real numbers). What is the sum of the possible values of MM?
SMT 2023 Discrete #5

Source:

5/3/2023
Ryan chooses five subsets S1,S2,S3,S4,S5S_1,S_2,S_3,S_4,S_5 of {1,2,3,4,5,6,7}\{1, 2, 3, 4, 5, 6, 7\} such that ∣S1∣=1|S_1| = 1, ∣S2∣=2|S_2| = 2, ∣S3∣=3|S_3| = 3, ∣S4∣=4|S_4| = 4, and ∣S5∣=5|S_5| = 5. Moreover, for all 1≤i<j≤51 \le i < j \le 5, either Si∩Sj=SiS_i \cap S_j = S_i or Si∩Sj=∅S_i \cap S_j = \emptyset (in other words, the intersection of SiS_i and SjS_j is either SiS_i or the empty set). In how many ways can Ryan select the sets?
SMT 2023 Geometry #5

Source:

8/9/2023
Equilateral triangle △ABC\vartriangle ABC has side length 1212 and equilateral triangles of side lengths a,b,c<6a, b, c < 6 are each cut from a vertex of △ABC\vartriangle ABC, leaving behind an equiangular hexagon A1A2B1B2C1C2A_1A_2B_1B_2C_1C_2, where A1A_1 lies on ACAC, A2A_2 lies on ABAB, and the rest of the vertices are similarly defined. Let A3A_3 be the midpoint of A1A2A_1A_2 and define B3B_3, C3C_3 similarly. Let the center of △ABC\vartriangle ABC be OO. Note that OA3OA_3, OB3OB_3, OC3OC_3 split the hexagon into three pentagons. If the sum of the areas of the equilateral triangles cut out is 18318\sqrt3 and the ratio of the areas of the pentagons is 5:6:75 : 6 : 7, what is the value of abcabc?
geometry