MathDB

3

Part of 2022 Stanford Mathematics Tournament

Problems(8)

SMT 2022 Algebra #3

Source:

3/27/2023
Compute ⌊112022+12023+⋯+12064⌋\left\lfloor\frac{1}{\frac{1}{2022}+\frac{1}{2023}+\dots+\frac{1}{2064}}\right\rfloor.
SMT 2022 Algebra Tiebreaker #3

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3/28/2023
Determine ⌊∏n=220222n+22n+1⌋,\left\lfloor\prod_{n=2}^{2022}\frac{2n+2}{2n+1}\right\rfloor, given that the answer is relatively prime to 20222022.
SMT 2022 Calculus #3

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3/29/2023
For k=1,2,…k=1,2,\dots, let fkf_k be the number of times sin⁡(kπx2)\sin\left(\frac{k\pi x}{2}\right) attains its maximum value on the interval x∈[0,1]x\in[0,1]. Compute lim⁡k→∞fkk.\lim_{k\rightarrow\infty}\frac{f_k}{k}.
SMT 2022 Calculus Tiebreaker #3

Source:

3/29/2023
Compute the value of ∫−ππex2−e−x2ex2−x2∣x∣dx.\int_{-\pi}^\pi\frac{e^{x^2}-e^{-x^2}}{e^{x^2}-x\sqrt{2}}|x|dx.
SMT 2022 Discrete #3

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4/1/2023
Every day you go to the music practice rooms at a random time between 12AM12\text{AM} and 8AM8\text{AM} and practice for 33 hours, while your friend goes at a random time from 5AM5\text{AM} to 11AM11\text{AM} and practices for 11 hour (the block of practice time need not be contained in he given time range for the arrival). What is the probability that you and your meet on at least 22 days in a given span of 55 days?
SMT 2022 Discrete Tiebreaker #3

Source:

4/1/2023
Five lilypads lie in a line on a pond. At first, a frog sits on the third lilypad. Then, each minute there is a 12\tfrac{1}{2} probability that the frog jumps to the lilypad to its left and 12\tfrac{1}{2} probability that it jumps to its right. If the frog jumps to the left from the leftmost lilypad or right from the rightmost lilypad, it will fall in the pond and stay there forever. Compute the probability that the frog is not in the pond after 1414 minutes have passed.
SMT 2022 Geometry #3

Source:

4/1/2023
△ABC\triangle ABC has side lengths 1313, 1414, and 1515. Let the feet of the altitudes from AA, BB, and CC be DD, EE, and FF, respectively. The circumcircle of △DEF\triangle DEF intersects ADAD, BEBE, and CFCF at II, JJ, and KK respectively. What is the area of △IJK\triangle IJK?
SMT 2022 Geometry Tiebreaker #3

Source:

4/1/2023
Let △ABC\triangle ABC be a triangle with BA<ACBA<AC, BC=10BC=10, and BA=8BA=8. Let HH be the orthocenter of △ABC\triangle ABC. Let FF be the point on segment ACAC such that BF=8BF=8. Let TT be the point of intersection of FHFH and the extension of line BCBC. Suppose that BT=8BT=8. Find the area of △ABC\triangle ABC.