MathDB

5

Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #5: Phi summation

Source:

5/30/2021
Let nn be the product of the first 1010 primes, and let S=∑xy∣nφ(x)⋅y,S=\sum_{xy\mid n} \varphi(x) \cdot y, where φ(x)\varphi(x) denotes the number of positive integers less than or equal to xx that are relatively prime to xx, and the sum is taken over ordered pairs (x,y)(x, y) of positive integers for which xyxy divides nn. Compute Sn.\tfrac{S}{n}.
Summationalgebranumber theory
2021 Combo #5: Bunny with changing die

Source:

5/30/2021
Teresa the bunny has a fair 88-sided die. Seven of its sides have fixed labels 1,2,⋯ ,7,1, 2, \cdots , 7, and the label on the eighth side can be changed and begins as 11. She rolls it several times, until each of 1,2,…,71, 2, \dots, 7 appears at least once. After each roll, if kk is the smallest positive integer that she has not rolled so far, she relabels the eighth side with kk. The probability that 77 is the last number she rolls is ab,\tfrac ab, where aa and bb are relatively prime positive integers. Compute 100a+b100a + b.
Combo
2021 Geo #5: Area Ratio

Source:

5/30/2021
Let AEFAEF be a triangle with EF=20EF = 20 and AE=AF=21AE = AF = 21. Let BB and DD be points chosen on segments AEAE and AF,AF, respectively, such that BDBD is parallel to EF.EF. Point CC is chosen in the interior of triangle AEFAEF such that ABCDABCD is cyclic. If BC=3BC = 3 and CD=4,CD = 4, then the ratio of areas [ABCD][AEF]\tfrac{[ABCD]}{[AEF]} can be written as ab\tfrac{a}{b} for relatively prime positive integers a,ba, b. Compute 100a+b100a + b.
geometryratio
2021 Team #5

Source:

6/27/2021
A convex polyhedron has nn faces that are all congruent triangles with angles 36∘,72∘36^{\circ}, 72^{\circ}, and 72∘72^{\circ}. Determine, with proof, the maximum possible value of nn.
3D geometrycombinatorics