MathDB

3

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

2023 Geometry #3

Source:

4/12/2023
Suppose ABCDABCD is a rectangle whose diagonals meet at EE. The perimeter of triangle ABEABE is 10π10\pi and the perimeter of triangle ADEADE is nn. Compute the number of possible integer values of nn.
HMMT Feb 2023 Team p3

Source:

2/20/2023
Let ABCDABCD be a convex quadrilateral such that ∠ABC=∠BCD=θ\angle ABC = \angle BCD = \theta for some acute angle θ\theta. Point XX lies inside the quadrilateral such that ∠XAD=∠XDA=90∘−θ\angle XAD = \angle XDA = 90^{\circ}-\theta. Prove that BX=XCBX = XC.
2023 Combinatorics #3

Source:

4/17/2023
Richard starts with the string HHMMMMTTHHMMMMTT. A move consists of replacing an instance of HMHM with MHMH, replacing an instance of MTMT with TMTM, or replacing an instance of THTH with HTHT. Compute the number of possible strings he can end up with after performing zero or more moves.
2023 Algebra/NT #3: Value of Trig Function

Source:

2/20/2023
Suppose xx is a real number such that sin⁡(1+cos⁡2x+sin⁡4x)=1314\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}. Compute cos⁡(1+sin⁡2x+cos⁡4x)\cos(1 + \sin^2 x + \cos^4 x).
trigonometryfunction