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2023 SMT Guts Round 2 p4-6 - Stanford Math Tournament

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August 31, 2023
geometryalgebranumber theoryStanford Math Tournament

Problem Statement

p4. For how many three-digit multiples of 1111 in the form abc‾\underline{abc} does the quadratic ax2+bx+cax^2 + bx + c have real roots?
p5. William draws a triangle △ABC\vartriangle ABC with AB=3AB =\sqrt3, BC=1BC = 1, and AC=2AC = 2 on a piece of paper and cuts out △ABC\vartriangle ABC. Let the angle bisector of ∠ABC\angle ABC meet ACAC at point DD. He folds △ABD\vartriangle ABD over BDBD. Denote the new location of point AA as A′A'. After William folds △A′CD\vartriangle A'CD over CDCD, what area of the resulting figure is covered by three layers of paper?
p6. Compute (1)(2)(3)+(2)(3)(4)+...+(18)(19)(20)(1)(2)(3) + (2)(3)(4) + ... + (18)(19)(20).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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