MathDB

7

Part of 2017 Harvard-MIT Mathematics Tournament

Problems(7)

2017 Algebra/NT #7: Largest number C satisfying Inequality

Source:

2/19/2017
Determine the largest real number cc such that for any 20172017 real numbers x1,x2,…,x2017x_1, x_2, \dots, x_{2017}, the inequality ∑i=12016xi(xi+xi+1)≥c⋅x20172\sum_{i=1}^{2016}x_i(x_i+x_{i+1})\ge c\cdot x^2_{2017} holds.
inequalities
2017 Team #7: Diverse Pascal's triangle row modulo p

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2/19/2017
Let pp be a prime. A complete residue class modulo pp is a set containing at least one element equivalent to k(modp)k \pmod{p} for all kk. (a) Show that there exists an nn such that the nnth row of Pascal's triangle forms a complete residue class modulo pp. (b) Show that there exists an n≤p2n \le p^2 such that the nnth row of Pascal's triangle forms a complete residue class modulo pp.
Pascal's Trianglenumber theory
2017 Geometry #7: Circumradius of AQR

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2/19/2017
Let ω\omega and Γ\Gamma be circles such that ω\omega is internally tangent to Γ\Gamma at a point PP. Let ABAB be a chord of Γ\Gamma tangent to ω\omega at a point QQ. Let R≠PR\neq P be the second intersection of line PQPQ with Γ\Gamma. If the radius of Γ\Gamma is 1717, the radius of ω\omega is 77, and AQBQ=3\frac{AQ}{BQ}=3, find the circumradius of triangle AQRAQR.
geometrycircumcircle
2017 Combinatorics #7: Frogs and Toads

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2/20/2017
There are 20172017 frogs and 20172017 toads in a room. Each frog is friends with exactly 22 distinct toads. Let NN be the number of ways to pair every frog with a toad who is its friend, so that no toad is paired with more than one frog. Let DD be the number of distinct possible values of NN, and let SS be the sum of all possible value of NN. Find the ordered pair (D,S)(D, S).
2017 Theme #7

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5/8/2018
On a blackboard a stranger writes the values of s7(n)2s_7(n)^2 for n=0,1,...,720−1n=0,1,...,7^{20}-1, where s7(n)s_7(n) denotes the sum of digits of nn in base 77. Compute the average value of all the numbers on the board.
combinatorics
2017 General #7

Source:

5/8/2018
Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?
combinatorics
2017 Guts #7: Neither set is a subset of the other

Source:

2/21/2017
An ordered pair of sets (A,B)(A, B) is good if AA is not a subset of BB and BB is not a subset of AA. How many ordered pairs of subsets of {1,2,…,2017}\{1, 2, \dots, 2017\} are good?
combinatorics