MathDB

10

Part of 2014 Harvard-MIT Mathematics Tournament

Problems(5)

2014 Algebra #10: Recursion for Number of Factors ≤ 9

Source:

7/7/2014
For an integer nn, let f9(n)f_9(n) denote the number of positive integers d≤9d\leq 9 dividing nn. Suppose that mm is a positive integer and b1,b2,…,bmb_1,b_2,\ldots,b_m are real numbers such that f9(n)=∑j=1mbjf9(n−j)f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j) for all n>mn>m. Find the smallest possible value of mm.
algebrapolynomialfunction
2014 Combinatorics #10: Up-right Path

Source:

2/23/2014
An up-right path from (a,b)∈R2(a, b) \in \mathbb{R}^2 to (c,d)∈R2(c, d) \in \mathbb{R}^2 is a finite sequence (x1,yz),…,(xk,yk)(x_1, y_z), \dots, (x_k, y_k) of points in R2 \mathbb{R}^2 such that (a,b)=(x1,y1),(c,d)=(xk,yk)(a, b)= (x_1, y_1), (c, d) = (x_k, y_k), and for each 1≤i<k1 \le i < k we have that either (xi+1,yy+1)=(xi+1,yi)(x_{i+1}, y_{y+1}) = (x_i+1, y_i) or (xi+1,yi+1)=(xi,yi+1)(x_{i+1}, y_{i+1}) = (x_i, y_i + 1). Two up-right paths are said to intersect if they share any point.
Find the number of pairs (A,B)(A, B) where AA is an up-right path from (0,0)(0, 0) to (4,4)(4, 4), BB is an up-right path from (2,0)(2, 0) to (6,4)(6, 4), and AA and BB do not intersect.
2014 Geometry #10: Another 13-14-15

Source:

2/25/2014
Let ABCABC be a triangle with AB=13AB = 13, BC=14BC = 14, and CA=15CA = 15. Let Γ\Gamma be the circumcircle of ABCABC, let OO be its circumcenter, and let MM be the midpoint of minor arc BCBC. Circle ω1\omega_1 is internally tangent to Γ\Gamma at AA, and circle ω2\omega_2, centered at MM, is externally tangent to ω1\omega_1 at a point TT. Ray ATAT meets segment BCBC at point SS, such that BS−CS=415BS - CS = \dfrac4{15}. Find the radius of ω2\omega_2
geometrycircumcirclegeometric transformationratioprojective geometry
2014 Guts #10: Subsets

Source:

2/25/2014
[6] Find the number of sets F\mathcal{F} of subsets of the set {1,…,2014}\{1,\ldots,2014\} such that:
a) For any subsets S1,S2∈F,S1∩S2∈FS_1,S_2 \in \mathcal{F}, S_1 \cap S_2 \in \mathcal{F}. b) If S∈FS \in \mathcal{F}, T⊆{1,…,2014}T \subseteq \{1,\ldots,2014\}, and S⊆TS \subseteq T, then T∈FT \in \mathcal{F}.
2014 Team #10: Complex Lower Bound for Final Number

Source:

3/2/2014
Fix a positive real number c>1c>1 and positive integer nn. Initially, a blackboard contains the numbers 1,c,…,cn−11,c,\ldots, c^{n-1}. Every minute, Bob chooses two numbers a,ba,b on the board and replaces them with ca+c2bca+c^2b. Prove that after n−1n-1 minutes, the blackboard contains a single number no less than (cn/L−1c1/L−1)L,\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L, where ϕ=1+52\phi=\tfrac{1+\sqrt 5}2 and L=1+log⁡ϕ(c)L=1+\log_\phi(c).
logarithmsfunctioncalculusderivative