MathDB

9

Part of 2023 Stanford Mathematics Tournament

Problems(3)

SMT 2023 Algebra #9

Source:

5/3/2023
Let x,y,zx,y,z be nonzero numbers, not necessarily real, such that (x−y)2+(y−z)2+(z−x)2=24yz(x-y)^2+(y-z)^2+(z-x)^2=24yz and x2yz+y2zx+z2xy=3.\tfrac{x^2}{yz}+\tfrac{y^2}{zx}+\tfrac{z^2}{xy}=3. Compute x2yz\tfrac{x^2}{yz}.
SMT 2023 Discrete #9

Source:

5/3/2023
Suppose aa and bb are positive integers with a curious property: (a3−3ab+12)n+(b3+12)n(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n is an integer for at least 33, but at most finitely many different choices of positive integers nn. What is the least possible value of a+ba+b?
SMT 2023 Geometry #9

Source:

8/9/2023
Triangle △ABC\vartriangle ABC is isosceles with AC=ABAC = AB, BC=1BC = 1, and ∠BAC=36o\angle BAC = 36^o. Let ω\omega be a circle with center B and radius rω=PABC4r_{\omega}= \frac{P_{ABC}}{4}, where PABCP_{ABC} denotes the perimeter of △ABC\vartriangle ABC. Let ω\omega intersect line ABAB at PP and line BCBC at QQ. Let IBI_B be the center of the excircle with of △ABC\vartriangle ABC with respect to point BB, and let BIBBI_B intersect PQP Q at SS. We draw a tangent line from SS to ⊙IB\odot I_B that intersects ⊙IB\odot I_B at point TT. Compute the length of ST.
geometry